"geometry of numbers" meaning in All languages combined

{
  "etymology_text": "The field was initiated by German mathematician Hermann Minkowski (1910, Geometrie der Zahlen).",
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      "expansion": "geometry of numbers (uncountable)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
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            "Entries with incorrect language header",
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        {
          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
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            "Formal sciences",
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            "All topics",
            "Fundamental"
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      "examples": [
        {
          "text": "A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in #x5C;textstyle#x5C;mathbb#x7B;R#x7D;ⁿ; the study of these lattices provides fundamental information on algebraic numbers.",
          "type": "example"
        },
        {
          "text": "1969, C. G. Lekkerkerker, Geometry of Numbers, Wolters-Noordoff, North-Holland, page 1,\nThe geometry of numbers to which this book is devoted deals with arbitrary bodies and arbitrary lattices in the n-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice."
        },
        {
          "ref": "2000, C. D. Olds, Anneli Lax, Giuliana Davidoff, The Geometry of Numbers, Mathematical Association of America:",
          "type": "quote"
        },
        {
          "text": "2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 181,\nThe easy proof is obtained applying the pigeon-hole principle to\nα₁x_1+…+αₙx_n(mod 1)|x_i=0,…N,\nor by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2ⁿ⁺¹ given by\n|X_0+α₁X_1+…αₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N."
        }
      ],
      "glosses": [
        "The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers."
      ],
      "id": "en-geometry_of_numbers-en-noun-G82Jrku9",
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          "number theory",
          "number theory"
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        [
          "geometry",
          "geometry"
        ],
        [
          "algebraic number",
          "algebraic number"
        ]
      ],
      "raw_glosses": [
        "(number theory) The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers."
      ],
      "tags": [
        "uncountable"
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
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      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "subbranch of number theory",
          "tags": [
            "feminine"
          ],
          "word": "Geometrie der Zahlen"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "subbranch of number theory",
          "tags": [
            "feminine"
          ],
          "word": "teoria dei numeri geometrica"
        }
      ],
      "wikipedia": [
        "Hermann Minkowski",
        "geometry of numbers"
      ]
    }
  ],
  "word": "geometry of numbers"
}

{
  "etymology_text": "The field was initiated by German mathematician Hermann Minkowski (1910, Geometrie der Zahlen).",
  "head_templates": [
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  "lang_code": "en",
  "pos": "noun",
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        "Terms with German translations",
        "Terms with Italian translations",
        "en:Number theory"
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      "examples": [
        {
          "text": "A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in #x5C;textstyle#x5C;mathbb#x7B;R#x7D;ⁿ; the study of these lattices provides fundamental information on algebraic numbers.",
          "type": "example"
        },
        {
          "text": "1969, C. G. Lekkerkerker, Geometry of Numbers, Wolters-Noordoff, North-Holland, page 1,\nThe geometry of numbers to which this book is devoted deals with arbitrary bodies and arbitrary lattices in the n-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice."
        },
        {
          "ref": "2000, C. D. Olds, Anneli Lax, Giuliana Davidoff, The Geometry of Numbers, Mathematical Association of America:",
          "type": "quote"
        },
        {
          "text": "2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 181,\nThe easy proof is obtained applying the pigeon-hole principle to\nα₁x_1+…+αₙx_n(mod 1)|x_i=0,…N,\nor by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2ⁿ⁺¹ given by\n|X_0+α₁X_1+…αₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N."
        }
      ],
      "glosses": [
        "The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers."
      ],
      "links": [
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          "number theory",
          "number theory"
        ],
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          "geometry",
          "geometry"
        ],
        [
          "algebraic number",
          "algebraic number"
        ]
      ],
      "raw_glosses": [
        "(number theory) The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers."
      ],
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        "uncountable"
      ],
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        "mathematics",
        "number-theory",
        "sciences"
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        "geometry of numbers"
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  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "subbranch of number theory",
      "tags": [
        "feminine"
      ],
      "word": "Geometrie der Zahlen"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "subbranch of number theory",
      "tags": [
        "feminine"
      ],
      "word": "teoria dei numeri geometrica"
    }
  ],
  "word": "geometry of numbers"
}