FAC

(algebraic geometry) Initialism of Faisceaux Algébriques Cohérents (a landmark paper by Jean-Pierre Serre)


rational

en-adj

rational (comparative more rational, superlative most rational)

more rational (comparative)
most rational (superlative)

From Old French rationel, rational, from Latin rationalis (“of or belonging to reason, rational, reasonable; having a ratio”), from ratio (“reason; calculation”).

der

/ˈɹæʃ(ə)nəl/

(mathematics, not comparable) Comprising, or expressible as, a ratio

(algebraic geometry) Of a variety: (informally) geometrically simple almost everywhere; (formally) birationally equivalent to projective space

(algebraic geometry) Of a function between varieties: acting as a morphism on an open subset of its domain.

(algebraic geometry) Of a point on an algebraic variety over a field: whose coordinates belong to the field in question (in contrast to those points of the variety which are only defined over the algebraic closure of the base field).


regular

en-adj

regular (comparative more regular, superlative most regular)

more regular (comparative)
most regular (superlative)

/ˈɹɛɡ.jʊ.lə/

/ˈɹɛɡ.jə.lɚ/

[ˈɹɛɡ.jɪ̈.lɚ]

From Middle English reguler, from Anglo-Norman reguler, Middle French reguler, regulier, and their source, Latin rēgulāris (“continuing rules for guidance”), from rēgula (“rule”), ultimately from Proto-Indo-European *reg- (“move in a straight line”).

مُنْتَظِمٌ

of a polygon: having all sides and corresponding angles equal

register

not etymologically related but has semantic associations

(algebraic geometry, not comparable, of a scheme) Such that the local ring at every point is regular.


tooth

Wales (usually), Midlands (especially Birmingham)

From Middle English tothe, toth, tooth, from Old English tōþ (“tooth”), from Proto-West Germanic *tanþ, from Proto-Germanic *tanþs (“tooth”), from Proto-Indo-European *h₃dónts (“tooth”). Related to tusk. Doublet of dent, dens, tind, and tine.
Cognates
Cognate with Scots tuth, tuith (“tooth”), North Frisian toth, tos (“tooth”), Dutch tand (“tooth”), German Zahn (“tooth”), Danish tand (“tooth”), Swedish tand (“tooth”), Norwegian tann (“tooth”), Icelandic tönn (“tooth”), Welsh dant (“tooth”), Latin dēns (“tooth”), Lithuanian dantìs (“tooth”), Ancient Greek ὀδούς (odoús)/ὀδών (odṓn, “tooth”), Armenian ատամ (atam), Persian دندان (dandân), Sanskrit दत् (dát), दन्त (danta, “tooth”).

(algebraic geometry) An irreducible component of a comb that intersects the handle in exactly one point, that point being distinct from the unique point of intersection for any other tooth of the comb.


curve

Attested since the 1690s, from Latin curvus (“bent, curved”), ultimately from Proto-Indo-European *(s)ker- (“to bend, curve, turn”) + *-wós. Doublet of curb, shrink, carcer, and cancer.

doublet

(algebraic geometry) An algebraic curve; a polynomial relation of the planar coordinates.


comb

From Middle English comb, from Old English camb (“comb”), from Proto-West Germanic *kamb, from Proto-Germanic *kambaz (“comb”), from Proto-Indo-European *ǵómbʰos (“tooth”), a doublet of cam. The verb is derived from the noun and displaced the older verb kemb.
Cognates
Compare Saterland Frisian Koum, Swedish/Dutch kam, Norwegian kam, German Kamm); also Tocharian B keme, Lithuanian žam̃bas (“sharp edge”), Old Church Slavonic зѫбъ (zǫbŭ), Albanian dhëmb, Ancient Greek γομφίος (gomphíos, “backtooth, molar”), Sanskrit जम्भ (jambha)).

cog

(algebraic geometry) A connected and reduced curve with irreducible components consisting of a smooth subcurve (called the handle) and one or more additional irreducible components (called teeth) that each intersect the handle in a single point that is unequal to the unique point of intersection for any of the other teeth.


flat

en-adj

flat (comparative flatter, superlative flattest)

flatter (comparative)
flattest (superlative)

bumpy

antonym(s) of “having no variations in altitude”

→ Portuguese: flat

From Middle English flat, a borrowing from Old Norse flatr (compare Norwegian and Swedish flat, Danish flad), from Proto-Germanic *flataz, from Proto-Indo-European *pleth₂- (“flat”); akin to Saterland Frisian flot (“smooth”), German Flöz (“a geological layer”), Ancient Greek πλατύς (platús), Latvian plats, Sanskrit प्रथस् (prathas, “extension”). Doublet of plat and pleyt.
The noun is from Middle English flat (“level piece of ground, flat edge of a weapon”), from the adjective.
The algebraic sense was coined by Serre in a 1956 paper, originally as French plat.

inh

(algebraic geometry, scheme theory, of a morphism of schemes) Such that the induced map on every stalk is flat (as a map of rings).


handle

desctree

→ Japanese: ハンドル (handoru)
→ Hokkien: 捍多路 (hăn-tó͘-luh)
→ Sakizaya: handulu

From Middle English handel, handle, from Old English handle (“a handle”), from handlian (“to handle, feel, deal with, discuss”). See verb below. Cognate with Danish handel (“a handle”).

(algebraic geometry) The smooth, irreducible subcurve of a comb which connects to each of the other components in exactly one point.


cubic

From Middle French cubique, equivalent to cube + -ic.


genus

en-noun

genus (plural genera or (all nonstandard) genuses or genusses or genii)

genera (plural)
genuses (nonstandard plural)
genusses (nonstandard plural)
genii (nonstandard plural)

→ Malayalam: ജീനസ് (jīnasŭ)

→ Swahili: jenasi

→ Thai: สกุล (sà-gun) (semantic loan)

Etymology tree
Proto-Indo-European *ǵenh₁-
Proto-Indo-European *-os
Proto-Indo-European *ǵénh₁os
Latin genusbor.
English genus
Borrowed from Latin genus (“birth, origin, a race, sort, kind”) from the root gen- in Latin gignō (“to beget, produce”). Doublet of gender, genre, and kin.

genus

rank in a taxonomic classification between family and species

(topology, graph theory, algebraic geometry) A natural number representing any of several related measures of the complexity of a given manifold or graph.


spectrum

From Latin spectrum (“appearance, image, apparition”), from speciō (“look at, view”). Doublet of specter. See also scope.

distinguished open sets
distinguished open sets

(commutative algebra, algebraic geometry) An abstract object in mathematics created from a commutative ring R and denoted operatorname Spec(R) or operatorname SpecR and said to be the spectrum of R; useful in the study of such rings for providing a geometric object which encodes many of the properties R, and in modern geometry for generalizing the notion of an algebraic variety to that of an affine scheme. Formally, the set of all prime ideals R equipped with the Zariski topology and augmented with a sheaf of rings called the structure sheaf, generated by the B-sheaf on the distinguished open sets D_f which assigns the localization of R at f to each set D_f, regarded as a ring of functions on D_f. See Spectrum of a ring on Wikipedia.Wikipedia


proper

en-adj

proper (comparative more proper, superlative most proper)

more proper (comparative)
most proper (superlative)

From Middle English propre, from Anglo-Norman proper, propre, Old French propre (French: propre), from Latin proprius.

(algebraic geometry, of a morphism of schemes) separated, of finite type, and universally closed.

(algebraic geometry, of a variety over a field k) such that unique morphism from the variety to k is proper (as above).


local

local (comparative more local, superlative most local)

more local (comparative)
most local (superlative)

From Middle English local, from Late Latin locālis (“belonging to a place”), possibly also via Old French local; ultimately from Latin locus (“a place”).
The ring-theoretic senses derive from Krull, who first referred to Noetherian commutative rings with a unique maximal ideal as "Stellenring" (Stellen (“place”) + ring) in 1938. The term was inspired by algebraic geometry, where local rings encode information about the behavior of curves (surfaces, etc.) at points; hence, describe "local" behavior.

der

Flatness is a local property.

(mathematics, not comparable, of a condition or property) Detectable from the behavior of substructures understood to be "near points;" in particular:

(algebra, algebraic geometry, of a property P of a ring R (or an R-module M)) Such that the following conditions are equivalent: (1) P holds for R (M); (2) P holds for the localization R_p (M_p) for all prime ideals p of R; (3) P holds for the localization R_m (M_m) for all maximal ideals m of R.


four-dimensional

(algebraic geometry) Having four dimensions; being measurable along four mutually perpendicular axes.


y-axis

independent variable
independent variable

(algebraic geometry) The axis on a graph that is usually drawn from bottom to top and usually shows the range of values of variable dependent on one other variable, or the second of two independent variables.


z-axis

(algebraic geometry) The axis on a graph of at least three dimensions that is usually drawn vertically and usually shows the range of values of a variable dependent on two other variables or the third independent variable.

z-as

axis on a graph that is usually drawn vertically


Cartesian coordinates

perpendicular distance
perpendicular distance

(algebraic geometry) The coordinates of a point measured as its perpendicular distance to the right of a vertical axis (the y-axis) and its perpendicular distance above a horizontal axis (the x-axis).


Cartesian distance

en-noun

Cartesian distance (plural Cartesian distances)

(algebraic geometry) Euclidean distance.


normal

en-adj

normal (comparative normaler or more normal, superlative normalest or most normal)

normaler (comparative)
more normal (comparative)
normalest (superlative)
most normal (superlative)

From Latin normālis (“made according to a carpenter's square; later: according to a rule”), from nōrma (“carpenter's square”), of uncertain origin; doublet of normale. The earliest meaning of the word in English was "perpendicular; forming a right angle" like something normālis (“made according to a carpenter's square”), but by Late Latin normālis had also come to mean "according to a rule", from which modern English senses of the word derive: in the 1800s, as people began to quantitatively study things like height, weight and blood pressure, the usual or most common values came to be called "normal", and by extension values regarded as healthy or desirable came to be called "normal" regardless of their usuality.

der

According to norms or rules or to a regular pattern.

(mathematics) Adhering to or being what is considered natural or regular in a particular field or context:

(algebraic geometry, of a variety or scheme) Such that the local ring at every point is an integrally closed domain.


nonic

(algebraic geometry) Describing a curve the equation of which is nonic.


variety

en-noun

variety (countable and uncountable, plural varieties)

From Middle French varieté (“variety”) (modern French variété (“variety; genre, type”)) or directly from its etymon Latin varietās (“difference; diversity, variety”) + English -ty (suffix forming abstract nouns from adjectives). Varietās is derived from varius (“different, diverse, various; variegated”) (possibly ultimately from Proto-Indo-European *h₁weh₂- (“to abandon; to give out; to leave”)) + -tās (suffix forming feminine abstract nouns indicating a state of being). The English word displaced the native Old English mislīcnes.
Sense 1.3.2 (“total number of distinct states of a system; logarithm to the base 2 of the total number of distinct states of a system”) was coined by the English psychiatrist William Ross Ashby (1903–1972) in his work An Introduction to Cybernetics (1956).
Cognates
* Galician variedade (“variety”)
* Italian varietà (“difference; variety”)
* Portuguese variedade (“variety”)
* Spanish variedad (“breed; variety”)

der

A collection or number of different things.

(algebraic geometry) Ellipsis of algebraic variety (“the set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition”).

algebraic variety
algebraic variety#English


geometry

en-noun

geometry (countable and uncountable, plural geometries)

From Middle English gemetry, geometrie, from Old French geometrie (modern French géométrie), from Latin geōmetria, from Ancient Greek γεωμετρία (geōmetría, “geometry, land-survey”), from γεωμέτρης (geōmétrēs, “land measurer”), from γῆ (gê, “earth, land, country”) + -μετρία (-metría, “measurement”), from μέτρον (métron, “a measure”). By surface analysis, geo- + -metry. Doublet of gematria.

der

(algebraic geometry, countable) A mathematical object comprising representations of a space and of its spatial relationships.


intercept

Borrowed from Latin interceptum, past participle of intercipiō.

the y intercept ― point at which a line crosses the y-axis

Because the horizontal-axis intercept occurs when y=0 and the vertical-axis intercept occurs when x=0, we can find the intercepts algebraically.

2012, Alice Kaseberg, Greg Cripe, Peter Wildman, Introductory Algebra: Everyday Explorations, page 278:

(algebraic geometry) The coordinate of the point at which a curve intersects an axis.


divisor

divisor

integer that divides another integer an integral number of times

(mathematics, algebraic geometry) Any of several kinds of formal object associated to an algebraic variety, scheme, etc., which play a role similar to that of homology or cohomology groups in the study of topological spaces.

(in the study of Riemann surfaces) An element of the free abelian group on the points of the space.

(on a variety (or integral locally Noetherian scheme)) A Weil divisor: an element of the free abelian group on the codimension-1 subvarieties (or subschemes).

(on a variety (or integral Noetherian scheme)) A Cartier divisor; see Cartier divisors on Wikipedia.Wikipedia


nef

Short for numerically effective; introduced by Miles Reid.

[…] this condition is the numerical consequence of the condition that for some mgt;0, the linear system #124;mD#124; is effective and free; thus nef = "numerically (effective and free)".

1983, Miles Reid, “Minimal Models of Canonical 3-Folds”, in Advanced Studies in Pure Mathematics, volume 1, page 131:

(algebraic geometry) Of a line bundle on a complete algebraic variety over a field: such that the degree of its restriction to every algebraic curve in the variety is non-negative.


toric

en-adj

toric (comparative more toric, superlative most toric)

more toric (comparative)
most toric (superlative)

Pertaining to or shaped like a torus, or a section of a torus; toroidal.

(geometry and algebra) Which, in any of several technical senses, admits a high degree of symmetry, allowing combinatorial methods to be used in its study.

(algebraic geometry, of a variety) Containing an algebraic torus as a dense subset, such that the group action of the torus on itself extends to the whole space; or, the embedding map taking the torus into the space. See Toric variety on Wikipedia.Wikipedia

(algebraic geometry, of a stack) Any of several generalizations of the notion of toric varieties to stacks: the stack quotient of a toric variety by its torus; the stack quotient of a toric variety by a subgroup of its torus.


irreducible

irreducible (comparative more irreducible, superlative most irreducible)

more irreducible (comparative)
most irreducible (superlative)

/ˌɪɹɪˈdjuːsɪbəl/

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(algebraic geometry, of an algebraic variety) Inexpressible as the union of two proper algebraic subvarieties.


adjoint

/ˈæd͡ʒ.ɔɪnt/

From French adjoindre (“to join”), from late 19th C; see also adjoin. Doublet of adjunct.
In the case of category theory (which brings together concepts from numerous fields), the term is often confounded with adjunct and the relationship is called an adjunction. The origin of any particular usage may therefore be uncertain.

(geometry, algebraic geometry) A curve A such that any point of a given curve C of multiplicity r has multiplicity at least r–1 on A. Sometimes the multiple points of C are required to be ordinary, and if this condition is not satisfied the term sub-adjoint is used.


algebraic function

en-noun

algebraic function (plural algebraic functions)

Quite often, algebraic functions are algebraic expressions using a finite number of terms and involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Some algebraic functions, however, cannot be expressed by such finite expressions.

The value of an algebraic function at an algebraic number is always an algebraic number.

1989, Manuel Bronstein, An Algorithm for the Integration of Elementary Functions, James H. Davenport (editor), EUROCAL '87: European Conference on Computer Algebra, Proceedings, Springer, LNCS 378, page 491,
Trager (1984) recently gave a new algorithm for the indefinite integration of algebraic functions.

1992, A. A. Goldberg, V. A. Pyana, Uniqueness Theorems for Algebraic Functions, A. B. Sossinsky (translation editor), Boris Yakovlevich Levin (editor), Entire and Subharmonic Functions, American Mathematical Society, page 199,
It is known that the algebraic function w=w(z) enables one to determine the polynomial P(z,w) up to a constant factor.

1993, Goro Kato (translator), Kenkichi Iwasawa, Algebraic Functions, American Mathematical Society, page xiv,
Riemann provided a clear picture of many difficult topics. His most significant contribution, rather than his construction of a Riemann surface from an algebraic function, was to derive the existence of algebraic functions as he built his theory based on the concept of a Riemann surface.

(algebraic geometry) Informally, any function expressible using (only) the operations of addition, subtraction, multiplication, division and raising to a rational power; more precisely, any continuous function definable as the root of some polynomial equation.


implicit function

implicit function (plural implicit functions)

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An important class of implicit functions comprises those defined by equations of the form x-f(y)#61;0. Choosing y as the value means that the implicit function is the inverse of f. This type of implicit function is of particular interest when that inverse is not expressible as a closed-form expression, and thus is difficult to study directly.

Thus, in the equation,
#92;quad#92;quad#92;quaday²#43;bxy#43;cx²#43;d#61;0;
y is an implicit function of x. Implicit functions are generally connected with their variables by one or more equations. When these equations are solved the implicit equation becomes explicit.
[…]The differential of an implicit function may be found without first finding the function itself.

1871, William G. Peck, Practical Treatise on the Differential and Integral Calculus, A. S. Barnes & Company, page 50:

The process of evaluating the derivatives in the case of implicit functions is exemplified below.

1875, J. Minot Rice, W. Woolsey Johnson, The Elements of the Differential Calculus, Part 2, Washington: Government Printing Office, page 71:

In this section we discuss the analysis of implicit functions for addressing questions like this.

2002, Michael W. Klein, Mathematical Methods for Economics, Addison-Wesley, page 245:

mathematical analysis
mathematical analysis

(mathematical analysis, algebraic geometry) A function defined by a (multivariable) implicit equation when one of the variables is regarded as the value of the function, especially where said equation is such that the value is not directly calculable from the other variables.

неявна функция

function defined as the solution to an implicit equation


rational function

rational function (plural rational functions)

Our first problem is that of interpolation in prescribed points to a given function by a rational function whose poles are given.

1960, J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 3rd edition, American Mathematical Society, page 184:

By Theorem 2.3.2., we have that the right-hand side of this equation can be equal to a rational function only if that rational function is equal to zero.

1970, Ellis Horowitz, Algorithms for Symbolic Integration of Rational Functions, University of Wisconsin-Madison, page 24:

Let #92;mathcal#123;C#125; be the class of continuous maps of #92;mathbb#123;C#125;#95;#92;infty into itself and let #92;mathcal#123;R#125; be the subclass of rational functions.[…]Now #92;mathcal#123;R#125; is a closed subset of #92;mathcal#123;C#125;#95;#92;infty because if the rational functions R#95;n converge uniformly to R on the complex sphere, then R is analytic on the sphere and so it too is rational.

2000, Alan F. Beardon, Iteration of Rational Functions: Complex Analytic Dynamical Systems, Springer, page 45:

(mathematics, complex analysis, algebraic geometry) Any function expressible as the quotient of two (coprime) polynomials (and which thus has poles at a finite, discrete set of points which are the roots of the denominator).

rationaalifunktio

function expressible as the quotient of polynomials


noetherian

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(algebraic geometry, of a scheme) admitting a finite covering by open affine subsets operatorname SpecA_i, where each A_i is a noetherian ring;


x-direction

(algebraic geometry) The direction aligned with the x-axis of a coordinate system.


y-direction

(algebraic geometry) The direction aligned with the y-axis of a coordinate system.


z-direction

(algebraic geometry) The direction aligned with the z-axis of a coordinate system.


cúbico

pt-adj

cúbico (feminine cúbica, masculine plural cúbicos, feminine plural cúbicas)

cúbica (feminine)
cúbicos (masculine plural)
cúbicas (feminine plural)

(algebraic geometry, of a polynomial) cubic (having an indeterminate to the power of 3i)


fourth dimension

(algebraic geometry) The fourth coordinate indicating a position along the fourth axis.


constructible

destructible

antonym(s) of “capable of being constructed”

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(algebraic geometry, of a subset of the spectrum of a noetherian ring) Admitting description as a finite union of locally closed (with respect to the Zariski topology) sets.


EGA

Éléments de géométrie algébrique
w:Éléments de géométrie algébrique

(algebraic geometry) Initialism of Éléments de géométrie algébrique.


isogeny

en-noun

isogeny (countable and uncountable, plural isogenies)

It is clear that if G#95;2 is connected, then two of the defining properties of an isogeny imply the third.

2000, Marc Hindry, Joseph H. Silverman, Diophantine Geometry: An Introduction, Springer, page 95:

2002, Mireille Fouquet, François Morain, Isogeny Volcanoes and the SEA Algorithm, Claus Fieker, David R. Kohel (editors), Algorithmic Number Theory: 5th International Symposium, Proceedings, Springer, LNCS 2369, page 279,
Lemma 2.2 Let E be an elliptic curve such that Z [π] is maximal at 𝓁. If there exists an 𝓁-isogeny of E, then this 𝓁-isogeny is an horizontal 𝓁-isogeny.

2005, Fred Diamond, Jerry Shurman, A First Course in Modular Forms, Springer, page 29,
The dual isogeny of an isomorphism is its inverse. The dual of a composition of isogenies is the composition of the duals in the reverse order. If φ is an isogeny and ̂φ is its dual then the formulas φ(z+Λ)=mz+Λ', φ(z'+Λ')=( operatorname φ/m)z'+Λ show that also
φ∘̂φ=[ operatorname deg(φ)]=[ operatorname deg(̂φ)],
so that φ is in turn the dual isogeny of its dual ̂φ. Isogeny of complex tori, rather than isomorphism, will turn out to be the appropriate equivalence relation in the context of modular forms.

(algebraic geometry, category theory) An epimorphism of group schemes that is surjective and has a finite kernel.


projectivization

Let us consider the vector space T#95;p. Its projectivization #123;PT#125;#95;p, is a projective space, which is obtained from T#95;p, by factorization relative to the collinearity of vectors. The projectivization #123;PT#125;#95;p is a set of straight lines, passing through the point p. Under projectivization, the cone C(2,r) becomes a manifold S(1,r-1) of dimension r.

1992, Maks A. Akivis, Alexander M. Shelekhov, translated by Vladislav V. Goldberg, Geometry and Algebra of Multidimensional Three-Webs, Springer, page 11:

1997, M. E. Alferieff (translator), Alexei Kostrikin, Yuri Manin, Linear Algebra and Geometry, Gordon and Breach Science Publishers, Paperback, page 233,
Therefore, the projectivization P(f) is determined only on the complement U_f=P(L)P( ker f).

The projectivization of a two-dimensional irregular equation (1.1) is a holomorphic vector field on #92;mathbbP¹#92;times#92;left#92;#123;#92;vertt#92;vertlt;1#92;right#92;#125; having a pair of singularities on the fiber #92;mathbbP¹#92;times 0 (which correspond to the eigenlines of the matrix A(0), the coordinate lines in our case).[…]Now the preceding corollary applied to the family of projectivizations says that the horizontal separatrices converge to the sectorial separatrices of the projectivized nonperturbed equation.

2004, Alexey Glutsyuk, “Confluence of singular points and Stokes phenomena”, in Yulij Ilyashenko, Christiane Rousseau, Gert Sabidussi, editors, Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Kluwer Academic, page 290:

(mathematics, algebraic geometry, birational geometry) A process (more formally, a mapping) that, given a vector space, specifies an associated projective space; (loosely) the projective space so specified.


squircle

At the NCI, the applicator shapes most often used are circular or "squircular." The squircle applicator has one square and one round end, and it is beveled so that the circular end is longer.

1985, Hospital Practice, volume 20, page 131:

(algebraic geometry) A closed quartic curve having properties intermediate between those of a square and a circle.

kwadratokrąg

shape intermediate between a square and a circle


cuatridimensional

es-adj

cuatridimensional m or f (masculine and feminine plural cuatridimensionales)

cuatridimensionales (feminine masculine plural)

(algebraic geometry, physics) four-dimensional


Zariski topology

en-noun

Zariski topology (plural Zariski topologies)

After Russian-born American mathematician Oscar Zariski (1899—1986), regarded as one of the most influential algebraic geometers of the 20th century.

The Zariski topology allows tools from topology to be used in the study of algebraic varieties, even when the underlying field is not a topological field.

Thus the Zariski open subsets are 'very big'; in particular, the Zariski topology is highly non-Hausdorff.
A further difference with the classical topology is that the Zariski topology on the product X#92;timesY of two affine varieties is stronger than the product of the Zariski topologies on X and Y.

1994, V. I. Danilov, “II. Algebraic Varieties and Schemes”, in I. R. Shafarevich, editor, Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes, Springer, page 184:

In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology. However, since the Zariski topology is so coarse, small neighbourhoods in the Zariski topology might not be local enough.

2008, Gert-Martin Greuel, Gerhard Pfister, A Singular Introduction to Commutative Algebra, 2nd edition, Springer, page 523:

If Y#92;subseteqKⁿ is an affine variety, then by definition the Zariski topology on Y has the subvarieties of Y as closed sets.

2011, Gregor Kemper, A Course in Commutative Algebra, Springer, page 34:

algebraic subvarieties
algebraic subvariety

(algebraic geometry) Originally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring.


elliptic curve

So called because it is the locus of an elliptic function.

(algebraic geometry) The algebraic set of an equation which equates the square of one variable to a reduced cubic polynomial in another variable, over some field, and such that the discriminant of the reduced cubic is non-zero.


affine variety

affine variety (plural affine varieties)

(algebraic geometry) A set of points (in n-dimensional space) which satisfy a set of equations which have a polynomial of n variables on one side and a zero on the other side.


K-theory

From circa 1960. The K stands for German Klasse (“class”).
The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

algebraic K-theory

study of rings generated by vector bundles

K'''-theory as an independent discipline is a fairly new subject, only about 35 years old.

1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

K'''-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K'''-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

topological K-theory
topological K-theory

(uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;

The study of rings R generated by the set of vector bundles over some topological space or scheme;
(dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.


anabelian

From an- (“not”) + abelian (“commutative”); coined to imply a nuancedly stronger condition than "merely" noncommutative (see an-).

We end up this section with the following remarks, both for the sake of completeness, but mostly because of the relevance of the facts we want to mention for the anabelian phenomena.

2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 522:

We will formulate Grothendieck's anabelian conjectures in a more general context later, after having presented the basic facts about ́etale fundamental groups.

2012, Florian Pop, “Lectures on anabelian phenomena in geometry and arithmetic”, in John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page 5:

But the interplay between those two parts remains mysterious and is at the source of some of the most standard conjectures about fundamental groups such as anabelian conjectures or the section conjecture.

2013, Anna Cadoret, “Galois Categories”, in Pierre Dèbes, Michel Emsalem, Matthieu Romagny, A. Muhammed Uludağ, editors, Arithmetic and Geometry Around Galois Theory, Springer (Birkhäuser), page 172:

algebraic fundamental group
algebraic fundamental group

(mathematics, algebraic geometry, arithmetic geometry) Of or relating to anabelian geometry, a proposed theory describing the way the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being an abelian group, in a sense to be made more precise.


PSL

special linear group
special linear group

(algebraic geometry) projective special linear group


eixo z

(algebraic geometry) z-axis (axis on a graph that is usually drawn as if vertical)


quadridimensional

pt-adj

quadridimensional m or f (plural quadridimensionais)


Zariski-Riemann space

en-noun

Zariski-Riemann space (plural Zariski-Riemann spaces)

Named by Nagata (1962) after Oscar Zariski and Bernhard Riemann.

(algebraic geometry) A locally ringed space of a subring k of a field K, whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve.

Bernhard Riemann
Oscar Zariski
Zariski-Riemann space


Iitaka dimension

Iitaka dimension (plural Iitaka dimensions)

(algebraic geometry) The Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L.


isocrystal

(algebraic geometry) A kind of p-adic analogue of a sheaf.


idirlíne

head

idirlíne f (genitive singular idirlíne, nominative plural idirlínte)

idirlíne (genitive singular)
idirlínte (nominative plural)
nominative (indefinite)
idirlíne (indefinite singular)
idirlínte (indefinite plural)
vocative (indefinite)
a idirlíne (indefinite singular)
a idirlínte (indefinite plural)
genitive (indefinite)
dative (indefinite)
nominative (definite)
an idirlíne (definite singular)
na hidirlínte (definite plural)
genitive (definite)
na hidirlíne (definite singular)
na n-idirlínte (definite plural)
dative (definite)
leis an idirlíne (definite singular)
don idirlíne (definite singular)
leis na hidirlínte (definite plural)
idirlíne
n-idirlíne
hidirlíne


idirlínigh

idirlínigh (present analytic idirlíníonn, future analytic idirlíneoidh, verbal noun idirlíniú, past participle idirlínithe)

idirlíníonn (analytic present)
idirlíneoidh (analytic future)
idirlíniú (noun-from-verb)
idirlínithe (participle past)
idirlíním (first-person indicative present singular)
idirlíníonn tú (indicative present second-person singular)
idirlínír (indicative present second-person singular)
idirlíníonn sé (indicative present singular third-person)
sí (indicative present singular third-person)
idirlínímid (first-person indicative plural present)
idirlíníonn muid (first-person indicative plural present)
idirlíníonn sibh (indicative plural present second-person)
idirlíníonn siad (indicative plural present third-person)
idirlíníd (indicative plural present third-person)
a idirlíníonn (indicative present relative)
a idirlíníos (indicative present relative)
a n-idirlíníonn (indicative present relative)
idirlínítear (autonomous indicative present)
d'idirlínigh mé (first-person indicative past singular)
d'idirlíníos (first-person indicative past singular)
idirlínigh mé‡ (first-person indicative past singular)
idirlíníos‡ (first-person indicative past singular)
d'idirlínigh tú (indicative past second-person singular)
d'idirlínís (indicative past second-person singular)
idirlínigh tú‡ (indicative past second-person singular)
idirlínís‡ (indicative past second-person singular)
d'idirlínigh sé (indicative past singular third-person)
sí (indicative past singular third-person)
idirlínigh sé (indicative past singular third-person)
sí‡ (indicative past singular third-person)
d'idirlíníomar (first-person indicative past plural)
d'idirlínigh muid (first-person indicative past plural)
idirlíníomar‡ (first-person indicative past plural)
idirlínigh muid‡ (first-person indicative past plural)
d'idirlínigh sibh (indicative past plural second-person)
d'idirlíníobhair (indicative past plural second-person)
idirlínigh sibh‡ (indicative past plural second-person)
idirlíníobhair‡ (indicative past plural second-person)
d'idirlínigh siad (indicative past plural third-person)
d'idirlíníodar (indicative past plural third-person)
idirlínigh siad‡ (indicative past plural third-person)
idirlíníodar‡ (indicative past plural third-person)
a d'idirlínigh (indicative past relative)
ar idirlínigh (indicative past relative)
idirlíníodh (autonomous indicative past)
hidirlíníodh (autonomous indicative past)
d'idirlínínn (first-person habitual indicative past singular)
idirlínínn‡ (first-person habitual indicative past singular)
n-idirlínínn‡‡ (first-person habitual indicative past singular)
d'idirlíníteá (habitual indicative past second-person singular)
idirlíníteá‡ (habitual indicative past second-person singular)
n-idirlíníteá‡‡ (habitual indicative past second-person singular)
d'idirlíníodh sé (habitual indicative past singular third-person)
sí (habitual indicative past singular third-person)
idirlíníodh sé (habitual indicative past singular third-person)
sí‡ (habitual indicative past singular third-person)
n-idirlíníodh sé (habitual indicative past singular third-person)
sí‡‡ (habitual indicative past singular third-person)
d'idirlínímis (first-person habitual indicative past plural)
d'idirlíníodh muid (first-person habitual indicative past plural)
idirlínímis‡ (first-person habitual indicative past plural)
idirlíníodh muid‡ (first-person habitual indicative past plural)
n-idirlínímis‡‡ (first-person habitual indicative past plural)
n-idirlíníodh muid‡‡ (first-person habitual indicative past plural)
d'idirlíníodh sibh (habitual indicative past plural second-person)
idirlíníodh sibh‡ (habitual indicative past plural second-person)
n-idirlíníodh sibh‡‡ (habitual indicative past plural second-person)
d'idirlínídís (habitual indicative past plural third-person)
d'idirlíníodh siad (habitual indicative past plural third-person)
idirlínídís (habitual indicative past plural third-person)
idirlíníodh siad‡ (habitual indicative past plural third-person)
n-idirlínídís‡‡ (habitual indicative past plural third-person)
n-idirlíníodh siad‡‡ (habitual indicative past plural third-person)
a d'idirlíníodh (habitual indicative past relative)
a n-idirlíníodh (habitual indicative past relative)
d'idirlínítí (autonomous habitual indicative past)
idirlínítí‡ (autonomous habitual indicative past)
n-idirlínítí‡‡ (autonomous habitual indicative past)
idirlíneoidh mé (first-person future singular)
idirlíneod (first-person future singular)
idirlíneochaidh mé (first-person future singular)
idirlíneoidh tú (future second-person singular)
idirlíneoir (future second-person singular)
idirlíneochaidh tú (future second-person singular)
idirlíneoidh sé (future singular third-person)
sí (future singular third-person)
idirlíneochaidh sé (future singular third-person)
idirlíneoimid (first-person future plural)
idirlíneoidh muid (first-person future plural)
idirlíneochaimid (first-person future plural)
idirlíneochaidh muid (first-person future plural)
idirlíneoidh sibh (future plural second-person)
idirlíneochaidh sibh (future plural second-person)
idirlíneoidh siad (future plural third-person)
idirlíneoid (future plural third-person)
idirlíneochaidh siad (future plural third-person)
a idirlíneoidh (future relative)
a idirlíneos (future relative)
a idirlíneochaidh (future relative)
a idirlíneochas (future relative)
a n-idirlíneoidh (future relative)
a n-idirlíneochaidh (future relative)
idirlíneofar (autonomous future)
idirlíneochar (autonomous future)
d'idirlíneoinn (conditional first-person singular)
d'idirlíneochainn (conditional first-person singular)
idirlíneoinn‡ (conditional first-person singular)
idirlíneochainn^†‡ (conditional first-person singular)
n-idirlíneoinn‡‡ (conditional first-person singular)
n-idirlíneochainn^†‡‡ (conditional first-person singular)
d'idirlíneofá (conditional second-person singular)
d'idirlíneochthá (conditional second-person singular)
idirlíneofá‡ (conditional second-person singular)
idirlíneochthá^†‡ (conditional second-person singular)
n-idirlíneofá‡‡ (conditional second-person singular)
n-idirlíneochthá^†‡‡ (conditional second-person singular)
d'idirlíneodh sé (conditional singular third-person)
sí (conditional singular third-person)
d'idirlíneochadh sé (conditional singular third-person)
idirlíneodh sé (conditional singular third-person)
sí‡ (conditional singular third-person)
idirlíneochadh sé (conditional singular third-person)
sí^†‡ (conditional singular third-person)
n-idirlíneodh sé (conditional singular third-person)
sí‡‡ (conditional singular third-person)
n-idirlíneochadh sé (conditional singular third-person)
sí^†‡‡ (conditional singular third-person)
d'idirlíneoimis (conditional first-person plural)
d'idirlíneodh muid (conditional first-person plural)
d'idirlíneochaimis (conditional first-person plural)
d'idirlíneochadh muid (conditional first-person plural)
idirlíneoimis‡ (conditional first-person plural)
idirlíneodh muid‡ (conditional first-person plural)
idirlíneochaimis^†‡ (conditional first-person plural)
idirlíneochadh muid^†‡ (conditional first-person plural)
n-idirlíneoimis‡‡ (conditional first-person plural)
n-idirlíneodh muid‡‡ (conditional first-person plural)
n-idirlíneochaimis^†‡‡ (conditional first-person plural)
n-idirlíneochadh muid^†‡‡ (conditional first-person plural)
d'idirlíneodh sibh (conditional plural second-person)
d'idirlíneochadh sibh (conditional plural second-person)
idirlíneodh sibh‡ (conditional plural second-person)
idirlíneochadh sibh^†‡ (conditional plural second-person)
n-idirlíneodh sibh‡‡ (conditional plural second-person)
n-idirlíneochadh sibh^†‡‡ (conditional plural second-person)
d'idirlíneoidís (conditional plural third-person)
d'idirlíneodh siad (conditional plural third-person)
d'idirlíneochadh siad (conditional plural third-person)
idirlíneoidís‡ (conditional plural third-person)
idirlíneodh siad‡ (conditional plural third-person)
idirlíneochadh siad^†‡ (conditional plural third-person)
n-idirlíneoidís‡‡ (conditional plural third-person)
n-idirlíneodh siad‡‡ (conditional plural third-person)
n-idirlíneochadh siad^†‡‡ (conditional plural third-person)
a d'idirlíneodh (conditional relative)
a d'idirlíneochadh (conditional relative)
a n-idirlíneodh (conditional relative)
a n-idirlíneochadh (conditional relative)
d'idirlíneofaí (autonomous conditional)
d'idirlíneochthaí (autonomous conditional)
idirlíneofaí‡ (autonomous conditional)
idirlíneochthaí^†‡ (autonomous conditional)
n-idirlíneofaí‡‡ (autonomous conditional)
n-idirlíneochthaí^†‡‡ (autonomous conditional)
go n-idirlíní mé (first-person present singular subjunctive)
go n-idirlíníod (first-person present singular subjunctive)
go n-idirlíní tú (present second-person singular subjunctive)
go n-idirlínír (present second-person singular subjunctive)
go n-idirlíní sé (present singular subjunctive third-person)
sí (present singular subjunctive third-person)
go n-idirlínímid (first-person plural present subjunctive)
go n-idirlíní muid (first-person plural present subjunctive)
go n-idirlíní sibh (plural present second-person subjunctive)
go n-idirlíní siad (plural present subjunctive third-person)
go n-idirlíníd (plural present subjunctive third-person)
go n-idirlínítear (autonomous present subjunctive)
dá n-idirlínínn (first-person past singular subjunctive)
dá n-idirlíníteá (past second-person singular subjunctive)
dá n-idirlíníodh sé (past singular subjunctive third-person)
sí (past singular subjunctive third-person)
dá n-idirlínímis (first-person past plural subjunctive)
dá n-idirlíníodh muid (first-person past plural subjunctive)
dá n-idirlíníodh sibh (past plural second-person subjunctive)
dá n-idirlínídís (past plural subjunctive third-person)
dá n-idirlíníodh siad (past plural subjunctive third-person)
dá n-idirlínítí (autonomous past subjunctive)
idirlíním (first-person imperative singular)
idirlínigh (imperative second-person singular)
idirlíníodh sé (imperative singular third-person)
sí (imperative singular third-person)
idirlínímis (first-person imperative plural)
idirlínígí (imperative plural second-person)
idirlínídh (imperative plural second-person)
idirlínídís (imperative plural third-person)
idirlínítear (autonomous imperative)
idirlíniú (noun-from-verb)
idirlínithe (participle past)
idirlínigh
n-idirlínigh
hidirlínigh

From idirlíne (“interline; intercept”, noun) + -igh (verbal suffix).

affix

(algebraic geometry, transitive) intercept


tetradimensional

gl-adj

tetradimensional m or f (plural tetradimensionais)


tetradimensional

es-adj

tetradimensional m or f (masculine and feminine plural tetradimensionales)

tetradimensionales (feminine masculine plural)


cuadridimensional

gl-adj

cuadridimensional m or f (plural cuadridimensionais)


cuadridimensional

es-adj

cuadridimensional m or f (masculine and feminine plural cuadridimensionales)

cuadridimensionales (feminine masculine plural)


subcomb

A corresponding series of descending subcombs lay in place, each successive subcomb's conduits, or teeth, having diameters decreased by half.

2007, Ron Sanders, Microcosmia, →ISBN, page 132:

With these choices, a subcomb with at least one tooth can be smoothed.

2013, Olivier Debarre, Higher-Dimensional Algebraic Geometry, →ISBN, page 106:

(algebraic geometry) A subcurve of a comb that includes the handle.


nefness

We also get some general results under nefness and relative semi-ampleness assumptions..

2016, Lei Zhang, “Subadditivity of Kodaira dimensions for fibrations from minimal models to projective varieties of general type”, in arXiv:

(algebraic geometry) The quality of being nef.


semiampleness

(algebraic geometry) The quality of being semiample.


semiample

(algebraic geometry) Having a positive multiple that is generated by global sections.


mirror symmetry

(algebraic geometry, theoretical physics) A relationship between Calabi-Yau manifolds where two such manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.


Grothendieck-Riemann-Roch theorem

Hirzebruch-Riemann-Roch theorem
Hirzebruch-Riemann-Roch theorem

(algebraic geometry) An important generalization of the Hirzebruch-Riemann-Roch theorem about complex manifolds.


algebraic variety

en-noun

algebraic variety (plural algebraic varieties)

If #92;mathcal#123;W#125; is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at x#92;in#92;mathcal#123;W#125; axe taken to be the germs of functions on #92;mathcal#123;W#125; induced by functions in the stalk #92;mathfrak#123;R#125;#95;x. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,#92;mathfrak#123;R#125;), by definition, is a local-ringed space.

2005, Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvili, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, page 87:

The quest for understanding the structure of algebraic varieties is a very natural one.[…]A landmark from that time was Hironaka's 1964 theorem on the existence of a smooth model for any complex algebraic variety.

2011, Carel Faber, Gerard van der Geer, Eduard Looijenga, Classification of Algebraic Varieties, European Mathematical Society, page vii:

Remark 2.8 Abstract algebraic varieties are obtained by gluing affine algebraic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3. This allows to develop the category of algebraic varieties with the usual constructions therein. All subsequent definitions could be formulated for abstract algebraic varieties, but will only be developed in the affine or quasi-affine case to keep things simple.

2014, Herwig Hauser, “Blowups and Resolution”, in David Ellwood, Herwig Hauser, Shigefumi Mori, Josef Schicho, editors, The Resolution of Singular Algebraic Varieties, American Mathematical Society, page 7:

(algebraic geometry) The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition.

varisto

set of solutions to a system of polynomial equations; any of certain generalisations of such a set

affine variety

set of solutions to a system of polynomial equations, any of certain generalisations of such a set


algebraic subvariety

algebraic subvariety (plural algebraic subvarieties)

(algebraic geometry) A subset of an algebraic variety that is itself an algebraic variety.


generic property

generic property (plural generic properties)

1989, Thomas S. Parker, Leon O. Chua, Appendix C: Differential Topology, Practical Numerical Algorithms for Chaotic Systems, Springer, Softcover reprint, page 314,
A property P that refers to members of a set Y is a generic property if the subset of Y whose members exhibit property P contains a dense open subset of Y.
Remark: If P₁ and P₂ are generic properties of a set Y, then so is the property "P₁ and P₂."

2001, Constantin I. Chueshov (translator), Igor Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, [1999, Russian edition] Acta Scientific Publishing House, page 117,
It should be noted that if a property of a dynamical system holds for the parameters from an open and dense set in the corresponding space, then it is frequently said that this property is a generic property.
However, it should be kept in mind that the generic property is not the one that holds almost always. […] Therefore, it should be remembered that generic properties are quite frequently encountered and stay stable during small perturbations of the properties of a system.

We now formulate the first generic property concerning the singularities of a Hamiltonian function.

2010, H. W. Broer, F. Takens, “Chapter 1: Preliminaries of Dynamical Systems Theory”, in B. Hasselblatt, H. W. Broer, F. Takens, editors, Handbook of Dynamical Systems, volume 3, Elsevier (North-Holland), page 24:

(topology, algebraic geometry) A property that is true in some dense open subset of a given set.


scheme theory

This book develops Grothendieck's scheme theory as a method for studying algebraic geometry.[…]In the preface of EGA [ Éléments de géométrie algébrique ], Grothendieck even claimed that a knowledge of classical algebraic geometry may hinder the reader from studying scheme theory.

1997, Kenji Ueno, translated by Goro Kato, Algebraic Geometry 1: From Algebraic Varieties to Schemes, American Mathematical Society, page x:

There is another approach dealing with more ring theoretical objects, i.e. starting from a nice class of algebras, called schematic algebras, it is possible to define a space, a Zariski topology, points, lines etc..., cf. [VOW1], [AZ], [ATV]. then it becomes possible to develop a scheme theory, completely extending the theory of schemes in classical algebraic geometry.

1999, Freddy Van Oystaeyen, “A Deformation of Projective Schemes”, in Freddy Van Oystaeyen, editor, Commutative Algebra and Algebraic Geometry, Marcel Dekker, page 295:

The main point of scheme theory is to consider C as a covariant functor R#92;mapstoC(R) from the category #92;textit#123;B-ALG#125; (also written as #92;textit#123;ALG#125;#95;#123;#47;#92;mathfrak#123;B

2004, Haruzo Hida, p-Adic Automorphic Forms on Shimura Varieties, Springer, page 68:

locally ringed space
locally ringed space

(algebraic geometry) The branch of mathematics that concerns schemes (locally ringed spaces admitting coverings by open sets, each isomorphic to the spectrum of some commutative ring).


projective variety

projective variety (plural projective varieties)

Varieties are sometimes called closed sets and some authors call an open subset of a projective variety a quasiprojective variety. The latter term is in an attempt to unify the concept of affine and projective variety.

2005, Max K. Agoston, Computer Graphics and Geometric Modelling: Mathematics, Springer, page 724:

A closed subset V#92;subset#92;mathbb#123;C#125;Pⁿ is called a (complex) projective variety if, locally, V is defined by a set of complex polynomial equations. Outside of its singular locus, that is, away from the subset where the defining equations do not have maximal rank, the projective variety is a complex submanifold of #92;mathbb#123;C#125;Pⁿ.

2006, Werner Ballmann, Lectures on Kähler Manifolds, European Mathematical Society, page 16:

The central topic of this note is a famous open problem to characterize which projective varieties are Kobayashi hyperbolic.

2015, Katsutoshi Yamanoi, “Kobayashi Hyperbolicity and Higher-dimensional Nevanlinna Theory”, in Takushiro Ochiai, Toshiki Mabuchi, Yoshiaki Maeda, Junjiro Noguchi, Alan Weinstein, editors, Geometry and Analysis on Manifolds: In Memory of Professor Shoshichi Kobayashi, Springer (Birkhäuser), page 209:

homogeneous polynomial
homogeneous polynomial

(algebraic geometry) A Zariski closed subvariety of a projective space; the zero-locus of a set of homogeneous polynomials that generates a prime ideal.


group functor

In order to be able to efficiently state the main results of this chapter (in §4) we formulate here a group-functor version of the R-nilpotent tower lemma (Ch.III, 6.4).

1972, A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations, Springer, page 106:

A direct product of k–group functors is again a k–group functor; so is a fibre product if the morphisms used in its construction are homomorphisms of k–group functors.

2003, Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd edition, American Mathematical Society, page 20:

By a group functor we mean a functor from small k-algebras to groups. A homomorphism of group functors is a natural transformation. A subgroup functor of a group functor G is a subfunctor H such that H(R) is a subgroup of G(R) for all R; it is normal if H(R) is normal in G(R) for all R.

2017, J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, page 118:

(category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.


cao độ

[kaːw˧˧ ʔɗo˧˨ʔ]

[kaːw˧˧ ʔɗow˨˩ʔ]

[kaːw˧˧ ʔɗow˨˩˨]

Sino-Vietnamese word from 高度 (“height; altitude”)

vi-etym-sino

(algebraic geometry) a z-coordinate; an applicate


algebraic group

en-noun

algebraic group (plural algebraic groups)

The purpose of this article is to discuss various results concerning the subgroups of simple algebraic groups G and of the corresponding finite groups of Lie type G^F (where F is a Frobenius morphism).

1998, Martin W. Liebeck, “Introduction to the subgroup structure of algebraic groups”, in Roger William Carter, Meinolf Geck, H. K. Moffatt, editors, Representations of Reductive Groups, Cambridge University Press, page 129:

We begin by showing how to make the quotient of an affine algebraic group by a normal algebraic subgroup into an affine algebraic group, and we determine the structure of commutative affine algebraic groups.

1998, Roe Goodman, Nolan R. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press, page 464:

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page 75,
We point out that some properties of an algebraic group are more easily read in its Lie algebra. An important property of an algebraic group is its solvability.

(algebraic geometry) A group that is an algebraic variety.


linear algebraic group

linear algebraic group (plural linear algebraic groups)

Geometric invariant theory arises in an attempt to construct a quotient of an algebraic variety X by an algebraic action of a linear algebraic group G.

2003, Igor Dolgachev, Lectures on Invariant Theory, Cambridge University Press, page xiii:

The differential Galois group of a homogeneous linear differential equation has a structure of linear algebraic group; hence it is endowed, in particular, with the Zariski topology.[…]Kolchin used the differential algebra developed by Ritt and also built the foundations of the theory of linear algebraic groups.

2011, Teresa Crespo, Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, American Mathematical Society, page xi:

2015, Willem A. de Graaf, Orbit Closures of Linear Algebraic Groups, Jaime Gutierrez, Josef Schicho, Martin Weimann (editors), Computer Algebra and Polynomials: Applications of Algebra and Number Theory, Springer, LNCS: 8942, page 76,
Actions of linear algebraic groups appear in many contexts. […] Throughout we assume that the base field is algebraically closed and of characteristic 0, as many constructions that we use (e.g., the correspondence between a linear algebraic group and its Lie algebra) only work well in characteristic 0.

general linear group
general linear group

(algebraic geometry, category theory) An algebraic group that is isomorphic to a subgroup of some general linear group.


regular map

Morphism of algebraic varieties
Regular map (graph theory)

We now find out how a regular map acts on the ring of regular functions on a closed set.[…]Moreover, regular maps can be characterised as the maps that take regular functions into regular functions.

1994, Igor R. Shafarevich, translated by Miles Reid, Basic Algebraic Geometry 1, 2nd edition, Springer, page 29:

Since one expects that the canonical projection f#58;X#92;rightarrowX#47;G is a regular map of algebraic varieties and so has closed fibres, all orbits must be closed subsets in the Zariski topology of X.

2017, José F. Fernando, José M. Gamboa, Carlos Ueno, Polynomial, regular and Nash images of Euclidean spaces, Fabrizio Broglia, Françoise Delon, Max Dickmann, Danielle Gondard-Cozette, Victoria Ann Powers (editors), Ordered Algebraic Structures and Related Topics: International Conference, American Mathematical Society, page 160,
The 1-dimensional semialgebraic set 𝒮:=x>0,xy=1 is the image of the regular map
f: R ²→ R ²,(x,y)↦((xy-1)²+x²,1/((xy-1)²+x²)).

(algebraic geometry) A morphism between algebraic varieties.


field extension

field extension (plural field extensions)

A Galois field extension may be defined in terms of its Galois group (Section 2) or in terms of the internal structure of the extension (Section 3).

1974, Thomas W. Hungerford, Algebra, Springer, page 230:

1998, David Goss, Basic Structures of Function Field Arithmetic, Springer, Corrected 2nd Printing, page 283,
Note that the extension of L obtained by adjoining all division points of ψ includes at most a finite constant field extension.

2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2bd Edition, page 530,
A field extension of a field K is, in particular, a K-algebra. Hence any two field extensions of K have a tensor product that is a K-algebra.

(algebra, field theory, algebraic geometry) Any pair of fields, denoted L/K, such that K is a subfield of L.

kuntalaajennus

pair of fields such that one is a subfield of the other


toạ độ Đề-các

head

toạ độ Đề-các

toạ độ Đề-các

vi-noun

toạ độ Đề-các

toạ độ Đề-các

[twaː˧˨ʔ ʔɗo˧˨ʔ ʔɗe˨˩ kaːk̚˧˦]

[twaː˨˩ʔ ʔɗow˨˩ʔ ʔɗej˦˩ kaːk̚˦˧˥]

[t⁽ʷ⁾aː˨˩˨ ʔɗow˨˩˨ ʔɗej˨˩ kaːk̚˦˥]

From toạ độ (“coordinates”) + French Descartes. Compare Mandarin 笛卡爾坐標 /笛卡尔坐标 (Díkǎ'ěr zuòbiāo, Địch Ca Nhĩ toạ tiêu), Japanese デカルト座標 (Dekaruto zahyō), Korean 데카르트 좌표 (Dekareuteu jwapyo).

cog

Cartesian coordinates
Cartesian coordinates#English

(collective, algebraic geometry) Cartesian coordinates


trục tung

[t͡ɕʊwk͡p̚˧˨ʔ tʊwŋ͡m˧˧]

[ʈʊwk͡p̚˨˩ʔ tʊwŋ͡m˧˧]

[ʈʊwk͡p̚˨˩˨ tʊwŋ͡m˧˧]

trục (“axis”) + Chinese 縱 /纵 (“vertical”)


tung độ

[tʊwŋ͡m˧˧ ʔɗo˧˨ʔ]

[tʊwŋ͡m˧˧ ʔɗow˨˩ʔ]

[tʊwŋ͡m˧˧ ʔɗow˨˩˨]

Sino-Vietnamese word from 縱 (“vertical”) and 度 (“degree”)

vi-etym-sino

(algebraic geometry) a y-coordinate; an ordinate


trục cao

[t͡ɕʊwk͡p̚˧˨ʔ kaːw˧˧]

[ʈʊwk͡p̚˨˩ʔ kaːw˧˧]

[ʈʊwk͡p̚˨˩˨ kaːw˧˧]


trục hoành

[t͡ɕʊwk͡p̚˧˨ʔ hwajŋ̟˨˩]

[ʈʊwk͡p̚˨˩ʔ hwɛɲ˦˩]

[ʈʊwk͡p̚˨˩˨ wan˨˩]

trục (“axis”) + Chinese 橫 /横 (“horizontal”)


hoành độ

[hwajŋ̟˨˩ ʔɗo˧˨ʔ]

[hwɛɲ˦˩ ʔɗow˨˩ʔ]

[wan˨˩ ʔɗow˨˩˨]

Sino-Vietnamese word from 橫 (“horizontal”) and 度 (“degree”)

(algebraic geometry) an x-coordinate; an abscissa


toạ độ

[twaː˧˨ʔ ʔɗo˧˨ʔ]

[twaː˨˩ʔ ʔɗow˨˩ʔ]

[t⁽ʷ⁾aː˨˩˨ ʔɗow˨˩˨]

Sino-Vietnamese word from 座 (“to sit”) and 度 (“degree”). Compare Chinese 坐標 /坐标 (toạ tiêu), Japanese 座標 (zahyō), Korean 좌표 (jwapyo).

(collective, algebraic geometry, geography) coordinates


デカルト座標

デカルト座標 (canonical)
Dekaruto zahyō (romanization)

From デカルト (Dekaruto, from French Descartes) + 座標 (zahyō, “coordinate”).

af

from French <i class="Latn mention" lang="fr">Descartes</i>

Cartesian coordinates
Cartesian coordinates

(algebraic geometry) Cartesian coordinates


varisto

varisto (nominative singular)
varistot (nominative plural)
varisto (accusative nominative singular)
varistot (accusative nominative plural)
variston (accusative genitive singular)
varistot (accusative genitive plural)
variston (genitive singular)
varistojen (genitive plural)
varistoa (partitive singular)
varistoja (partitive plural)
varistossa (inessive singular)
varistoissa (inessive plural)
varistosta (elative singular)
varistoista (elative plural)
varistoon (illative singular)
varistoihin (illative plural)
varistolla (adessive singular)
varistoilla (adessive plural)
varistolta (ablative singular)
varistoilta (ablative plural)
varistolle (allative singular)
varistoille (allative plural)
varistona (essive singular)
varistoina (essive plural)
varistoksi (singular translative)
varistoiksi (plural translative)
varistotta (abessive singular)
varistoitta (abessive plural)
varistoin (instructive plural)
See the possessive forms below. (comitative)
varistoni (nominative singular)
varistoni (nominative plural)
varistoni (accusative nominative singular)
varistoni (accusative nominative plural)
varistoni (accusative genitive singular)
varistoni (accusative genitive plural)
varistoni (genitive singular)
varistojeni (genitive plural)
varistoani (partitive singular)
varistojani (partitive plural)
varistossani (inessive singular)
varistoissani (inessive plural)
varistostani (elative singular)
varistoistani (elative plural)
varistooni (illative singular)
varistoihini (illative plural)
varistollani (adessive singular)
varistoillani (adessive plural)
varistoltani (ablative singular)
varistoiltani (ablative plural)
varistolleni (allative singular)
varistoilleni (allative plural)
varistonani (essive singular)
varistoinani (essive plural)
varistokseni (singular translative)
varistoikseni (plural translative)
varistottani (abessive singular)
varistoittani (abessive plural)
varistoineni (comitative plural)
varistosi (nominative singular)
varistosi (nominative plural)
varistosi (accusative nominative singular)
varistosi (accusative nominative plural)
varistosi (accusative genitive singular)
varistosi (accusative genitive plural)
varistosi (genitive singular)
varistojesi (genitive plural)
varistoasi (partitive singular)
varistojasi (partitive plural)
varistossasi (inessive singular)
varistoissasi (inessive plural)
varistostasi (elative singular)
varistoistasi (elative plural)
varistoosi (illative singular)
varistoihisi (illative plural)
varistollasi (adessive singular)
varistoillasi (adessive plural)
varistoltasi (ablative singular)
varistoiltasi (ablative plural)
varistollesi (allative singular)
varistoillesi (allative plural)
varistonasi (essive singular)
varistoinasi (essive plural)
varistoksesi (singular translative)
varistoiksesi (plural translative)
varistottasi (abessive singular)
varistoittasi (abessive plural)
varistoinesi (comitative plural)
varistomme (nominative singular)
varistomme (nominative plural)
varistomme (accusative nominative singular)
varistomme (accusative nominative plural)
varistomme (accusative genitive singular)
varistomme (accusative genitive plural)
varistomme (genitive singular)
varistojemme (genitive plural)
varistoamme (partitive singular)
varistojamme (partitive plural)
varistossamme (inessive singular)
varistoissamme (inessive plural)
varistostamme (elative singular)
varistoistamme (elative plural)
varistoomme (illative singular)
varistoihimme (illative plural)
varistollamme (adessive singular)
varistoillamme (adessive plural)
varistoltamme (ablative singular)
varistoiltamme (ablative plural)
varistollemme (allative singular)
varistoillemme (allative plural)
varistonamme (essive singular)
varistoinamme (essive plural)
varistoksemme (singular translative)
varistoiksemme (plural translative)
varistottamme (abessive singular)
varistoittamme (abessive plural)
varistoinemme (comitative plural)
varistonne (nominative singular)
varistonne (nominative plural)
varistonne (accusative nominative singular)
varistonne (accusative nominative plural)
varistonne (accusative genitive singular)
varistonne (accusative genitive plural)
varistonne (genitive singular)
varistojenne (genitive plural)
varistoanne (partitive singular)
varistojanne (partitive plural)
varistossanne (inessive singular)
varistoissanne (inessive plural)
varistostanne (elative singular)
varistoistanne (elative plural)
varistoonne (illative singular)
varistoihinne (illative plural)
varistollanne (adessive singular)
varistoillanne (adessive plural)
varistoltanne (ablative singular)
varistoiltanne (ablative plural)
varistollenne (allative singular)
varistoillenne (allative plural)
varistonanne (essive singular)
varistoinanne (essive plural)
varistoksenne (singular translative)
varistoiksenne (plural translative)
varistottanne (abessive singular)
varistoittanne (abessive plural)
varistoinenne (comitative plural)
varistonsa (nominative singular)
varistonsa (nominative plural)
varistonsa (accusative nominative singular)
varistonsa (accusative nominative plural)
varistonsa (accusative genitive singular)
varistonsa (accusative genitive plural)
varistonsa (genitive singular)
varistojensa (genitive plural)
varistoaan (partitive singular)
varistoansa (partitive singular)
varistojaan (partitive plural)
varistojansa (partitive plural)
varistossaan (inessive singular)
varistossansa (inessive singular)
varistoissaan (inessive plural)
varistoissansa (inessive plural)
varistostaan (elative singular)
varistostansa (elative singular)
varistoistaan (elative plural)
varistoistansa (elative plural)
varistoonsa (illative singular)
varistoihinsa (illative plural)
varistollaan (adessive singular)
varistollansa (adessive singular)
varistoillaan (adessive plural)
varistoillansa (adessive plural)
varistoltaan (ablative singular)
varistoltansa (ablative singular)
varistoiltaan (ablative plural)
varistoiltansa (ablative plural)
varistolleen (allative singular)
varistollensa (allative singular)
varistoilleen (allative plural)
varistoillensa (allative plural)
varistonaan (essive singular)
varistonansa (essive singular)
varistoinaan (essive plural)
varistoinansa (essive plural)
varistokseen (singular translative)
varistoksensa (singular translative)
varistoikseen (plural translative)
varistoiksensa (plural translative)
varistottaan (abessive singular)
varistottansa (abessive singular)
varistoittaan (abessive plural)
varistoittansa (abessive plural)
varistoineen (comitative plural)
varistoinensa (comitative plural)

Possibly a blend of English variety + monisto. Contains the suffix -sto.

algebrallinen varisto


x-suunta

x-suunta (nominative singular)
x-suunnat (nominative plural)
x-suunta (accusative nominative singular)
x-suunnat (accusative nominative plural)
x-suunnan (accusative genitive singular)
x-suunnat (accusative genitive plural)
x-suunnan (genitive singular)
x-suuntien (genitive plural)
x-suuntain (genitive plural rare)
x-suuntaa (partitive singular)
x-suuntia (partitive plural)
x-suunnassa (inessive singular)
x-suunnissa (inessive plural)
x-suunnasta (elative singular)
x-suunnista (elative plural)
x-suuntaan (illative singular)
x-suuntiin (illative plural)
x-suunnalla (adessive singular)
x-suunnilla (adessive plural)
x-suunnalta (ablative singular)
x-suunnilta (ablative plural)
x-suunnalle (allative singular)
x-suunnille (allative plural)
x-suuntana (essive singular)
x-suuntina (essive plural)
x-suunnaksi (singular translative)
x-suunniksi (plural translative)
x-suunnatta (abessive singular)
x-suunnitta (abessive plural)
x-suunnin (instructive plural)
See the possessive forms below. (comitative)
x-suuntani (nominative singular)
x-suuntani (nominative plural)
x-suuntani (accusative nominative singular)
x-suuntani (accusative nominative plural)
x-suuntani (accusative genitive singular)
x-suuntani (accusative genitive plural)
x-suuntani (genitive singular)
x-suuntieni (genitive plural)
x-suuntaini (genitive plural rare)
x-suuntaani (partitive singular)
x-suuntiani (partitive plural)
x-suunnassani (inessive singular)
x-suunnissani (inessive plural)
x-suunnastani (elative singular)
x-suunnistani (elative plural)
x-suuntaani (illative singular)
x-suuntiini (illative plural)
x-suunnallani (adessive singular)
x-suunnillani (adessive plural)
x-suunnaltani (ablative singular)
x-suunniltani (ablative plural)
x-suunnalleni (allative singular)
x-suunnilleni (allative plural)
x-suuntanani (essive singular)
x-suuntinani (essive plural)
x-suunnakseni (singular translative)
x-suunnikseni (plural translative)
x-suunnattani (abessive singular)
x-suunnittani (abessive plural)
x-suuntineni (comitative plural)
x-suuntasi (nominative singular)
x-suuntasi (nominative plural)
x-suuntasi (accusative nominative singular)
x-suuntasi (accusative nominative plural)
x-suuntasi (accusative genitive singular)
x-suuntasi (accusative genitive plural)
x-suuntasi (genitive singular)
x-suuntiesi (genitive plural)
x-suuntaisi (genitive plural rare)
x-suuntaasi (partitive singular)
x-suuntiasi (partitive plural)
x-suunnassasi (inessive singular)
x-suunnissasi (inessive plural)
x-suunnastasi (elative singular)
x-suunnistasi (elative plural)
x-suuntaasi (illative singular)
x-suuntiisi (illative plural)
x-suunnallasi (adessive singular)
x-suunnillasi (adessive plural)
x-suunnaltasi (ablative singular)
x-suunniltasi (ablative plural)
x-suunnallesi (allative singular)
x-suunnillesi (allative plural)
x-suuntanasi (essive singular)
x-suuntinasi (essive plural)
x-suunnaksesi (singular translative)
x-suunniksesi (plural translative)
x-suunnattasi (abessive singular)
x-suunnittasi (abessive plural)
x-suuntinesi (comitative plural)
x-suuntamme (nominative singular)
x-suuntamme (nominative plural)
x-suuntamme (accusative nominative singular)
x-suuntamme (accusative nominative plural)
x-suuntamme (accusative genitive singular)
x-suuntamme (accusative genitive plural)
x-suuntamme (genitive singular)
x-suuntiemme (genitive plural)
x-suuntaimme (genitive plural rare)
x-suuntaamme (partitive singular)
x-suuntiamme (partitive plural)
x-suunnassamme (inessive singular)
x-suunnissamme (inessive plural)
x-suunnastamme (elative singular)
x-suunnistamme (elative plural)
x-suuntaamme (illative singular)
x-suuntiimme (illative plural)
x-suunnallamme (adessive singular)
x-suunnillamme (adessive plural)
x-suunnaltamme (ablative singular)
x-suunniltamme (ablative plural)
x-suunnallemme (allative singular)
x-suunnillemme (allative plural)
x-suuntanamme (essive singular)
x-suuntinamme (essive plural)
x-suunnaksemme (singular translative)
x-suunniksemme (plural translative)
x-suunnattamme (abessive singular)
x-suunnittamme (abessive plural)
x-suuntinemme (comitative plural)
x-suuntanne (nominative singular)
x-suuntanne (nominative plural)
x-suuntanne (accusative nominative singular)
x-suuntanne (accusative nominative plural)
x-suuntanne (accusative genitive singular)
x-suuntanne (accusative genitive plural)
x-suuntanne (genitive singular)
x-suuntienne (genitive plural)
x-suuntainne (genitive plural rare)
x-suuntaanne (partitive singular)
x-suuntianne (partitive plural)
x-suunnassanne (inessive singular)
x-suunnissanne (inessive plural)
x-suunnastanne (elative singular)
x-suunnistanne (elative plural)
x-suuntaanne (illative singular)
x-suuntiinne (illative plural)
x-suunnallanne (adessive singular)
x-suunnillanne (adessive plural)
x-suunnaltanne (ablative singular)
x-suunniltanne (ablative plural)
x-suunnallenne (allative singular)
x-suunnillenne (allative plural)
x-suuntananne (essive singular)
x-suuntinanne (essive plural)
x-suunnaksenne (singular translative)
x-suunniksenne (plural translative)
x-suunnattanne (abessive singular)
x-suunnittanne (abessive plural)
x-suuntinenne (comitative plural)
x-suuntansa (nominative singular)
x-suuntansa (nominative plural)
x-suuntansa (accusative nominative singular)
x-suuntansa (accusative nominative plural)
x-suuntansa (accusative genitive singular)
x-suuntansa (accusative genitive plural)
x-suuntansa (genitive singular)
x-suuntiensa (genitive plural)
x-suuntainsa (genitive plural rare)
x-suuntaansa (partitive singular)
x-suuntiaan (partitive plural)
x-suuntiansa (partitive plural)
x-suunnassaan (inessive singular)
x-suunnassansa (inessive singular)
x-suunnissaan (inessive plural)
x-suunnissansa (inessive plural)
x-suunnastaan (elative singular)
x-suunnastansa (elative singular)
x-suunnistaan (elative plural)
x-suunnistansa (elative plural)
x-suuntaansa (illative singular)
x-suuntiinsa (illative plural)
x-suunnallaan (adessive singular)
x-suunnallansa (adessive singular)
x-suunnillaan (adessive plural)
x-suunnillansa (adessive plural)
x-suunnaltaan (ablative singular)
x-suunnaltansa (ablative singular)
x-suunniltaan (ablative plural)
x-suunniltansa (ablative plural)
x-suunnalleen (allative singular)
x-suunnallensa (allative singular)
x-suunnilleen (allative plural)
x-suunnillensa (allative plural)
x-suuntanaan (essive singular)
x-suuntanansa (essive singular)
x-suuntinaan (essive plural)
x-suuntinansa (essive plural)
x-suunnakseen (singular translative)
x-suunnaksensa (singular translative)
x-suunnikseen (plural translative)
x-suunniksensa (plural translative)
x-suunnattaan (abessive singular)
x-suunnattansa (abessive singular)
x-suunnittaan (abessive plural)
x-suunnittansa (abessive plural)
x-suuntineen (comitative plural)
x-suuntinensa (comitative plural)

[ˈæks̠ˌs̠uːn̪t̪ɑ̝]


y-suunta

y-suunta (nominative singular)
y-suunnat (nominative plural)
y-suunta (accusative nominative singular)
y-suunnat (accusative nominative plural)
y-suunnan (accusative genitive singular)
y-suunnat (accusative genitive plural)
y-suunnan (genitive singular)
y-suuntien (genitive plural)
y-suuntain (genitive plural rare)
y-suuntaa (partitive singular)
y-suuntia (partitive plural)
y-suunnassa (inessive singular)
y-suunnissa (inessive plural)
y-suunnasta (elative singular)
y-suunnista (elative plural)
y-suuntaan (illative singular)
y-suuntiin (illative plural)
y-suunnalla (adessive singular)
y-suunnilla (adessive plural)
y-suunnalta (ablative singular)
y-suunnilta (ablative plural)
y-suunnalle (allative singular)
y-suunnille (allative plural)
y-suuntana (essive singular)
y-suuntina (essive plural)
y-suunnaksi (singular translative)
y-suunniksi (plural translative)
y-suunnatta (abessive singular)
y-suunnitta (abessive plural)
y-suunnin (instructive plural)
See the possessive forms below. (comitative)
y-suuntani (nominative singular)
y-suuntani (nominative plural)
y-suuntani (accusative nominative singular)
y-suuntani (accusative nominative plural)
y-suuntani (accusative genitive singular)
y-suuntani (accusative genitive plural)
y-suuntani (genitive singular)
y-suuntieni (genitive plural)
y-suuntaini (genitive plural rare)
y-suuntaani (partitive singular)
y-suuntiani (partitive plural)
y-suunnassani (inessive singular)
y-suunnissani (inessive plural)
y-suunnastani (elative singular)
y-suunnistani (elative plural)
y-suuntaani (illative singular)
y-suuntiini (illative plural)
y-suunnallani (adessive singular)
y-suunnillani (adessive plural)
y-suunnaltani (ablative singular)
y-suunniltani (ablative plural)
y-suunnalleni (allative singular)
y-suunnilleni (allative plural)
y-suuntanani (essive singular)
y-suuntinani (essive plural)
y-suunnakseni (singular translative)
y-suunnikseni (plural translative)
y-suunnattani (abessive singular)
y-suunnittani (abessive plural)
y-suuntineni (comitative plural)
y-suuntasi (nominative singular)
y-suuntasi (nominative plural)
y-suuntasi (accusative nominative singular)
y-suuntasi (accusative nominative plural)
y-suuntasi (accusative genitive singular)
y-suuntasi (accusative genitive plural)
y-suuntasi (genitive singular)
y-suuntiesi (genitive plural)
y-suuntaisi (genitive plural rare)
y-suuntaasi (partitive singular)
y-suuntiasi (partitive plural)
y-suunnassasi (inessive singular)
y-suunnissasi (inessive plural)
y-suunnastasi (elative singular)
y-suunnistasi (elative plural)
y-suuntaasi (illative singular)
y-suuntiisi (illative plural)
y-suunnallasi (adessive singular)
y-suunnillasi (adessive plural)
y-suunnaltasi (ablative singular)
y-suunniltasi (ablative plural)
y-suunnallesi (allative singular)
y-suunnillesi (allative plural)
y-suuntanasi (essive singular)
y-suuntinasi (essive plural)
y-suunnaksesi (singular translative)
y-suunniksesi (plural translative)
y-suunnattasi (abessive singular)
y-suunnittasi (abessive plural)
y-suuntinesi (comitative plural)
y-suuntamme (nominative singular)
y-suuntamme (nominative plural)
y-suuntamme (accusative nominative singular)
y-suuntamme (accusative nominative plural)
y-suuntamme (accusative genitive singular)
y-suuntamme (accusative genitive plural)
y-suuntamme (genitive singular)
y-suuntiemme (genitive plural)
y-suuntaimme (genitive plural rare)
y-suuntaamme (partitive singular)
y-suuntiamme (partitive plural)
y-suunnassamme (inessive singular)
y-suunnissamme (inessive plural)
y-suunnastamme (elative singular)
y-suunnistamme (elative plural)
y-suuntaamme (illative singular)
y-suuntiimme (illative plural)
y-suunnallamme (adessive singular)
y-suunnillamme (adessive plural)
y-suunnaltamme (ablative singular)
y-suunniltamme (ablative plural)
y-suunnallemme (allative singular)
y-suunnillemme (allative plural)
y-suuntanamme (essive singular)
y-suuntinamme (essive plural)
y-suunnaksemme (singular translative)
y-suunniksemme (plural translative)
y-suunnattamme (abessive singular)
y-suunnittamme (abessive plural)
y-suuntinemme (comitative plural)
y-suuntanne (nominative singular)
y-suuntanne (nominative plural)
y-suuntanne (accusative nominative singular)
y-suuntanne (accusative nominative plural)
y-suuntanne (accusative genitive singular)
y-suuntanne (accusative genitive plural)
y-suuntanne (genitive singular)
y-suuntienne (genitive plural)
y-suuntainne (genitive plural rare)
y-suuntaanne (partitive singular)
y-suuntianne (partitive plural)
y-suunnassanne (inessive singular)
y-suunnissanne (inessive plural)
y-suunnastanne (elative singular)
y-suunnistanne (elative plural)
y-suuntaanne (illative singular)
y-suuntiinne (illative plural)
y-suunnallanne (adessive singular)
y-suunnillanne (adessive plural)
y-suunnaltanne (ablative singular)
y-suunniltanne (ablative plural)
y-suunnallenne (allative singular)
y-suunnillenne (allative plural)
y-suuntananne (essive singular)
y-suuntinanne (essive plural)
y-suunnaksenne (singular translative)
y-suunniksenne (plural translative)
y-suunnattanne (abessive singular)
y-suunnittanne (abessive plural)
y-suuntinenne (comitative plural)
y-suuntansa (nominative singular)
y-suuntansa (nominative plural)
y-suuntansa (accusative nominative singular)
y-suuntansa (accusative nominative plural)
y-suuntansa (accusative genitive singular)
y-suuntansa (accusative genitive plural)
y-suuntansa (genitive singular)
y-suuntiensa (genitive plural)
y-suuntainsa (genitive plural rare)
y-suuntaansa (partitive singular)
y-suuntiaan (partitive plural)
y-suuntiansa (partitive plural)
y-suunnassaan (inessive singular)
y-suunnassansa (inessive singular)
y-suunnissaan (inessive plural)
y-suunnissansa (inessive plural)
y-suunnastaan (elative singular)
y-suunnastansa (elative singular)
y-suunnistaan (elative plural)
y-suunnistansa (elative plural)
y-suuntaansa (illative singular)
y-suuntiinsa (illative plural)
y-suunnallaan (adessive singular)
y-suunnallansa (adessive singular)
y-suunnillaan (adessive plural)
y-suunnillansa (adessive plural)
y-suunnaltaan (ablative singular)
y-suunnaltansa (ablative singular)
y-suunniltaan (ablative plural)
y-suunniltansa (ablative plural)
y-suunnalleen (allative singular)
y-suunnallensa (allative singular)
y-suunnilleen (allative plural)
y-suunnillensa (allative plural)
y-suuntanaan (essive singular)
y-suuntanansa (essive singular)
y-suuntinaan (essive plural)
y-suuntinansa (essive plural)
y-suunnakseen (singular translative)
y-suunnaksensa (singular translative)
y-suunnikseen (plural translative)
y-suunniksensa (plural translative)
y-suunnattaan (abessive singular)
y-suunnattansa (abessive singular)
y-suunnittaan (abessive plural)
y-suunnittansa (abessive plural)
y-suuntineen (comitative plural)
y-suuntinensa (comitative plural)

[ˈyːˌs̠uːn̪t̪ɑ̝]


z-suunta

z-suunta (nominative singular)
z-suunnat (nominative plural)
z-suunta (accusative nominative singular)
z-suunnat (accusative nominative plural)
z-suunnan (accusative genitive singular)
z-suunnat (accusative genitive plural)
z-suunnan (genitive singular)
z-suuntien (genitive plural)
z-suuntain (genitive plural rare)
z-suuntaa (partitive singular)
z-suuntia (partitive plural)
z-suunnassa (inessive singular)
z-suunnissa (inessive plural)
z-suunnasta (elative singular)
z-suunnista (elative plural)
z-suuntaan (illative singular)
z-suuntiin (illative plural)
z-suunnalla (adessive singular)
z-suunnilla (adessive plural)
z-suunnalta (ablative singular)
z-suunnilta (ablative plural)
z-suunnalle (allative singular)
z-suunnille (allative plural)
z-suuntana (essive singular)
z-suuntina (essive plural)
z-suunnaksi (singular translative)
z-suunniksi (plural translative)
z-suunnatta (abessive singular)
z-suunnitta (abessive plural)
z-suunnin (instructive plural)
See the possessive forms below. (comitative)
z-suuntani (nominative singular)
z-suuntani (nominative plural)
z-suuntani (accusative nominative singular)
z-suuntani (accusative nominative plural)
z-suuntani (accusative genitive singular)
z-suuntani (accusative genitive plural)
z-suuntani (genitive singular)
z-suuntieni (genitive plural)
z-suuntaini (genitive plural rare)
z-suuntaani (partitive singular)
z-suuntiani (partitive plural)
z-suunnassani (inessive singular)
z-suunnissani (inessive plural)
z-suunnastani (elative singular)
z-suunnistani (elative plural)
z-suuntaani (illative singular)
z-suuntiini (illative plural)
z-suunnallani (adessive singular)
z-suunnillani (adessive plural)
z-suunnaltani (ablative singular)
z-suunniltani (ablative plural)
z-suunnalleni (allative singular)
z-suunnilleni (allative plural)
z-suuntanani (essive singular)
z-suuntinani (essive plural)
z-suunnakseni (singular translative)
z-suunnikseni (plural translative)
z-suunnattani (abessive singular)
z-suunnittani (abessive plural)
z-suuntineni (comitative plural)
z-suuntasi (nominative singular)
z-suuntasi (nominative plural)
z-suuntasi (accusative nominative singular)
z-suuntasi (accusative nominative plural)
z-suuntasi (accusative genitive singular)
z-suuntasi (accusative genitive plural)
z-suuntasi (genitive singular)
z-suuntiesi (genitive plural)
z-suuntaisi (genitive plural rare)
z-suuntaasi (partitive singular)
z-suuntiasi (partitive plural)
z-suunnassasi (inessive singular)
z-suunnissasi (inessive plural)
z-suunnastasi (elative singular)
z-suunnistasi (elative plural)
z-suuntaasi (illative singular)
z-suuntiisi (illative plural)
z-suunnallasi (adessive singular)
z-suunnillasi (adessive plural)
z-suunnaltasi (ablative singular)
z-suunniltasi (ablative plural)
z-suunnallesi (allative singular)
z-suunnillesi (allative plural)
z-suuntanasi (essive singular)
z-suuntinasi (essive plural)
z-suunnaksesi (singular translative)
z-suunniksesi (plural translative)
z-suunnattasi (abessive singular)
z-suunnittasi (abessive plural)
z-suuntinesi (comitative plural)
z-suuntamme (nominative singular)
z-suuntamme (nominative plural)
z-suuntamme (accusative nominative singular)
z-suuntamme (accusative nominative plural)
z-suuntamme (accusative genitive singular)
z-suuntamme (accusative genitive plural)
z-suuntamme (genitive singular)
z-suuntiemme (genitive plural)
z-suuntaimme (genitive plural rare)
z-suuntaamme (partitive singular)
z-suuntiamme (partitive plural)
z-suunnassamme (inessive singular)
z-suunnissamme (inessive plural)
z-suunnastamme (elative singular)
z-suunnistamme (elative plural)
z-suuntaamme (illative singular)
z-suuntiimme (illative plural)
z-suunnallamme (adessive singular)
z-suunnillamme (adessive plural)
z-suunnaltamme (ablative singular)
z-suunniltamme (ablative plural)
z-suunnallemme (allative singular)
z-suunnillemme (allative plural)
z-suuntanamme (essive singular)
z-suuntinamme (essive plural)
z-suunnaksemme (singular translative)
z-suunniksemme (plural translative)
z-suunnattamme (abessive singular)
z-suunnittamme (abessive plural)
z-suuntinemme (comitative plural)
z-suuntanne (nominative singular)
z-suuntanne (nominative plural)
z-suuntanne (accusative nominative singular)
z-suuntanne (accusative nominative plural)
z-suuntanne (accusative genitive singular)
z-suuntanne (accusative genitive plural)
z-suuntanne (genitive singular)
z-suuntienne (genitive plural)
z-suuntainne (genitive plural rare)
z-suuntaanne (partitive singular)
z-suuntianne (partitive plural)
z-suunnassanne (inessive singular)
z-suunnissanne (inessive plural)
z-suunnastanne (elative singular)
z-suunnistanne (elative plural)
z-suuntaanne (illative singular)
z-suuntiinne (illative plural)
z-suunnallanne (adessive singular)
z-suunnillanne (adessive plural)
z-suunnaltanne (ablative singular)
z-suunniltanne (ablative plural)
z-suunnallenne (allative singular)
z-suunnillenne (allative plural)
z-suuntananne (essive singular)
z-suuntinanne (essive plural)
z-suunnaksenne (singular translative)
z-suunniksenne (plural translative)
z-suunnattanne (abessive singular)
z-suunnittanne (abessive plural)
z-suuntinenne (comitative plural)
z-suuntansa (nominative singular)
z-suuntansa (nominative plural)
z-suuntansa (accusative nominative singular)
z-suuntansa (accusative nominative plural)
z-suuntansa (accusative genitive singular)
z-suuntansa (accusative genitive plural)
z-suuntansa (genitive singular)
z-suuntiensa (genitive plural)
z-suuntainsa (genitive plural rare)
z-suuntaansa (partitive singular)
z-suuntiaan (partitive plural)
z-suuntiansa (partitive plural)
z-suunnassaan (inessive singular)
z-suunnassansa (inessive singular)
z-suunnissaan (inessive plural)
z-suunnissansa (inessive plural)
z-suunnastaan (elative singular)
z-suunnastansa (elative singular)
z-suunnistaan (elative plural)
z-suunnistansa (elative plural)
z-suuntaansa (illative singular)
z-suuntiinsa (illative plural)
z-suunnallaan (adessive singular)
z-suunnallansa (adessive singular)
z-suunnillaan (adessive plural)
z-suunnillansa (adessive plural)
z-suunnaltaan (ablative singular)
z-suunnaltansa (ablative singular)
z-suunniltaan (ablative plural)
z-suunniltansa (ablative plural)
z-suunnalleen (allative singular)
z-suunnallensa (allative singular)
z-suunnilleen (allative plural)
z-suunnillensa (allative plural)
z-suuntanaan (essive singular)
z-suuntanansa (essive singular)
z-suuntinaan (essive plural)
z-suuntinansa (essive plural)
z-suunnakseen (singular translative)
z-suunnaksensa (singular translative)
z-suunnikseen (plural translative)
z-suunniksensa (plural translative)
z-suunnattaan (abessive singular)
z-suunnattansa (abessive singular)
z-suunnittaan (abessive plural)
z-suunnittansa (abessive plural)
z-suuntineen (comitative plural)
z-suuntinensa (comitative plural)

[ˈts̠e̞t̪ɑ̝ˌs̠uːn̪t̪ɑ̝]

[ˈts̠e̞tˌs̠uːn̪t̪ɑ̝]


vanishing ideal

en-noun

vanishing ideal (plural vanishing ideals)

(algebraic geometry) An ideal consisting of all polynomials (belonging to a certain polynomial ring) which zero out within a given algebraic variety.


birational geometry

Birational geometry amounts to the study of mappings given by rational functions rather than polynomials.

1999, Yongbin Ruan, Surgery, Quantum Cohomology and Birational Geometry, Ya. Eliashberg, D. Fuchs, T. Ratiu, Alan Weinstein (editors), Northern California Symplectic Geometry Seminar, American Mathematical Society, Translations, Series 2, Volume 196, page 183,
Recently, some amazing relations between quantum cohomology and birational geometry have been discovered.

2008, Paltin Ionescu, Birational geometry of rationally connected manifolds via quasi-lines, Ciro Ciliberto, et al., Projective Varieties with Unexpected Properties, Walter de Gruyter, page 317,
This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ℙⁿ (quasi-lines).

2009, Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez, Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, Springer (Birkhäuser), Progress in Mathematics 276, page 233,
In this chapter we offer some applications of Fourier-Mukai transforms, namely, a classification of the Fourier-Mukai partners of complex projective surfaces, some issues in birational geometry, and an approach to the McKay correspondence via Fourier-Mukai transform.

(algebraic geometry) A field of algebraic geometry in which the aim is to determine under what conditions two algebraic varieties are isomorphic outside lower-dimensional subsets.


algebraic analysis

The area of study in algebraic geometry began as a research programme, instigated in 1959 by Japanese mathematician Mikio Sato.

1986, Goro Kato (translator), Masaki Kashiwara, Takahiro Kawai, Tatsuo Kimura, Foundations of Algebraic Analysis, Princeton Universty Press, https://books.google.com.au/books?id=qYUrDgAAQBAJ&printsec=frontcover&dq=%22algebraic+analysis%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwiK5MSlvsLsAhWXbn0KHbPuBPQQ6AEwAnoECAYQAg#v=onepage&q=%22algebraic%20analysis%22&f=false.

In their classical work [123], Kashiwara, Kawai, and Kimura pointedly observe that the Japanese algebraic analysis is really the algebraic analysis in the tradition of Euler; we may add that what we mean by algebraic microlocal analysis is the successful attempt to adapt the methods of abstract algebraic geometry to the non-commutative setting in which the base ring is now the ring of variable coefficients^([sic]) partial differential operators.

1999, Goro Kato, Daniele C. Struppa, Fundamentals of Algebraic Microlocal Analysis, Marcel Dekker, page v:

2008, Yoshitsugu Takei, The work of T. Kawai on exact WKB analysis, T. Aoki, H. Majima, Y. Takei, N. Tose (editors), Algebraic Analysis of Differential Equations, Springer, page 19,
Thus Kawai has been engaged mainly in the research of exact WKB analysis or algebraic analysis of singular perturbation theory since 1989.

partial differential equation
partial differential equation

(algebraic geometry, algebraic topology) The study of systems of linear partial differential equations, using techniques from sheaf theory and complex analysis to examine functions and certain generalisations of functions (such as hyperfunctions and microfunctions).


algebraic K-theory

Algebraic cycles are typically studied by imposing one of several equivalence relations. The equivalence relation most relevant for algebraic K-theory is rational equivalence.

1999, E. M. Friedlander, “Lecture VII. Beilinson's vision”, in H. Bass, A. O. Kuku, C. Pedrini, editors, Algebraic K-theory And Its Applications, World Scientific, page 61:

Algebraic K-theory has two components: the classical theory which centers around the Grothendieck group K#95;0 of a category and uses explicit algebraic presentations and higher algebraic K-theory which requires topological or homological machinery to define.

2013, Charles A. Weibel, The K-book: An Introduction to Algebraic K-theory, American Mathematical Society, page ix:

2014, Daniel Scott Farley, Ivonne Johanna Ortiz, Algebraic K-theory of Crystallographic Groups, Springer, Lecture Notes in Mathematics 2113, page 1,
Algebraic K-theory is a branch of algebra dealing with linear algebra over a general ring R instead of a field. […] Algebraic K-theory plays an important part in many areas of mathematics, especially number theory, algebraic topology and algebraic geometry.

(algebraic geometry) K-theory studied from the point of view of algebra.

K-théorie algébrique

K theory studied from the point of view of algebra


Witt group

Named after German mathematician Ernst Witt (1911–1991), who introduced the concept in 1937.

groupe de Witt

group of equivalence classes found within a field, variety or category

equivalence relation
equivalence relation

metabolic quadratic space
metabolic quadratic space

(algebra) Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;

Given a field k of characteristic ≠ 2, the abelian group of equivalence classes of nondegenerate symmetric bilinear forms over k (where the equivalence relation is such that two forms are equivalent if each is obtainable from the other by adding a metabolic quadratic space), with the group operation corresponding to that of orthogonal direct sum of forms;
(algebraic geometry, by extension) given a variety X, the quotient of the Grothendieck group of isometry classes of quadratic spaces on X, with respect to orthogonal sum, modulo the subgroup generated by metabolic spaces
(category theory, by extension) given a category with duality, the quotient of isometry classes of symmetric spaces, modulo metabolic spaces.


algebraic fundamental group

algebraic fundamental group (plural algebraic fundamental groups)

1997, Frans Oort, The Algebraic Fundamental Group, Leila Schneps, Pierre Lochak (editors), Geometric Galois Actions 1: Around Grothendieck's Esquisse D'un Programme, Cambridge University Press, London Mathematical Society, page 78,
Serre constructed an example of an algebraic variety X over a number field K plus two embeddings of K into ℂ such that the geometric fundamental groups of X_1( ℂ) and X_2( ℂ) are not isomorphic, while the algebraic fundamental groups (i.e. their profinite completions) clearly are isomorphic, see (Serre).

Neves and Papadakis ([NP09]) constructed a numerical Campedelli surface with algebraic fundamental group #92;mathbbZ#95;6, while Lee and Park ([LP09]) constructed one with algebraic fundamental group #92;mathbbZ#95;2, and one with algebraic fundamental group #92;mathbbZ#95;3 was added in the second version of the same paper.

2011, Ingrid Bauer, Fabrizio Catanese, Roberto Pignatelli, “Surfaces of general type with geometric genus zero: a survey”, in Wolfgang Ebeling, Klaus Hulek, Knut Smoczyk, editors, Complex and Differential Geometry, Springer, page 11:

In order to work in arbitrary characteristic, we should use the algebraic fundamental group, which we denote by #92;pi#95;1. (A very good introduction to algebraic fundamental groups is [24].[…])

2013, János Kollár, “Rationally Connected Varieties and Connected Groups”, in Károly Böröczky, Jr., János Kollár, Szamuely Tamas, editors, Higher Dimensional Varieties and Rational Points, Springer, page 70:

(algebraic geometry) A group which is an analogue for schemes of the fundamental group for topological spaces.


anabelian geometry

anabelian geometry (countable and uncountable, plural anabelian geometries)

We shall say more about the noncommutative nature of anabelian geometries after we state the main theorem (Theorem 1.17 below) on these geometries.

1999, Shinichi Mochizuki, Foundations of p-adic Teichmüller Theory, American Mathematical Society, page 63:

The basic idea of Grothendieck's anabelian geometry is that under certain "anabelian hypotheses" the étale fundamental group of a scheme contains all the geometric and arithmetic information about the scheme in discussion, that is to say, the scheme is functorially encoded in its étale fundamental group.
Among these we shall discuss applications to the inverse Galois problem, Belyi's theorem on covers of the projective line minus three points and some advanced results on ‘anabelian geometry’ of curves.

2000, Florian Pop, “Alterations and Birational Anabelian Geometry”, in H. Hauser, J. Lipman, F. Oort, A. Quirós, editors, Resolution of Singularities, Birkhäuser, page 519:

Therefore, the time seems right to encourage a much broader understanding of the arithmetic issues surrounding anabelian geometry and its ramifications.
While the overall importance of the theorems of anabelian geometry appears to be widely acknowledged, there is as yet not much specific knowledge within the arithmetic geometry community of its coherent body of concepts and philosophy, and of the new technology that yields actual results.

2012, John Coates, Minhyong Kim, Florian Pop, Mohamed Saïdi, Peter Schneider, editors, Non-abelian Fundamental Groups and Iwasawa Theory, Cambridge University Press, page vii:

(mathematics, algebraic geometry, arithmetic geometry) A theory which describes the way in which the algebraic fundamental group G of an algebraic variety (or some related geometric object) V determines how V can be mapped into another geometric object W, under the assumption that G is very far from being abelian (commutative).


racionala funkcio

eo-head

racionala funkcio (accusative singular racionalan funkcion, plural racionalaj funkcioj, accusative plural racionalajn funkciojn)

racionalan funkcion (accusative singular)
racionalaj funkcioj (plural)
racionalajn funkciojn (accusative plural)

rational function
rational function#English

(mathematics, complex analysis, algebraic geometry) rational function


Białynicki-Birula decomposition

en-noun

Białynicki-Birula decomposition (plural Białynicki-Birula decompositions)

The description depends on a partial ordering of the fixed point components which arises from the Białynicki-Birula decompositions of X.

1984, Pacific Journal of Mathematics, volumes 114-115, page 149:

Section 2 describes a C*-action on Grassmann manifolds and gives the corresponding Białynicki-Birula decomposition.

1988, Mathematica Scandinavica, volume 62, page 5:

(algebraic geometry) A T-stable cellular decomposition of a smooth T-variety.


Sumihiro's theorem

(algebraic geometry) A theorem stating that a normal algebraic variety with an action of a torus can be covered by torus-invariant affine open subsets.


隱函數

/in²¹⁴⁻²¹ xän³⁵ ʂu⁵¹/

/in²¹⁴⁻²¹ xän³⁵ ʂu⁵¹/

(mathematical analysis, algebraic geometry) implicit function


Donaldson-Thomas theory

Donaldson-Thomas invariant
Donaldson-Thomas invariant

(algebraic geometry) The theory of Donaldson-Thomas invariants.


Donaldson-Thomas invariant

Donaldson-Thomas invariant (plural Donaldson-Thomas invariants)

Introduced in 1998 by Simon Donaldson and Richard Thomas.

Calabi-Yau threefold
Calabi-Yau manifold#English

(algebraic geometry) Given a compact moduli space of sheaves on a Calabi-Yau threefold, its Donaldson-Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class.


Calabi-Yau manifold

Calabi-Yau manifold (plural Calabi-Yau manifolds)

The name was coined by Candelas et al. (1985), after Eugenio Calabi (1954, 1957) who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978), who proved the Calabi conjecture.

(algebraic geometry) A kind of manifold having properties, such as Ricci flatness, that yield applications in theoretical physics.


橢圓曲線

ㄊㄨㄛˇ ㄩㄢˊ ㄑㄩ ㄒㄧㄢˋ

тоюань цюйсянь

/tʰu̯ɔ²¹⁴⁻²¹ ɥɛn³⁵ t͡ɕʰy⁵⁵ ɕi̯ɛn⁵¹/

/tʰɔː¹³ jyːn²¹ kʰʊk̚⁵ siːn³³/

/tʰu̯ɔ²¹⁴⁻²¹ ɥɛn³⁵ t͡ɕʰy⁵⁵ ɕi̯ɛn⁵¹/

/tʰɔː¹³ jyːn²¹ kʰʊk̚⁵ siːn³³/