geometry of numbers

== English == === Etymology === The field was initiated by German mathematician Hermann Minkowski (1910, Geometrie der Zahlen). === Noun === geometry of numbers (uncountable) (number theory) The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers. 1969, C. G. Lekkerkerker, Geometry of Numbers, Wolters-Noordoff, North-Holland, page 1, The geometry of numbers to which this book is devoted deals with arbitrary bodies and arbitrary lattices in the n {\displaystyle n} -dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice. 2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Cambridge University Press, page 181, The easy proof is obtained applying the pigeon-hole principle to { α 1 x 1 + ⋯ + α n x n ( mod 1 ) | x i = 0 , … N } {\displaystyle \{\alpha _{1}x_{1}+\dots +\alpha _{n}x_{n}{\pmod {1}}\vert x_{i}=0,\dots N\}} , or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2 n + 1 {\displaystyle 2^{n+1}} given by | X 0 + α 1 X 1 + … α n X n | ≤ N − n , | X i | ≤ N , i = 1 , … N {\displaystyle \vert X_{0}+\alpha _{1}X_{1}+\dots \alpha _{n}X_{n}\vert \leq N^{-n},\vert X_{i}\vert \leq N,i=1,\dots N} . ==== Translations ==== === Further reading === Category:Geometry of numbers on Wikipedia.Wikipedia