Geometry of numbers

English

Etymology

The field was initiated by German mathematician Template:W (1910, Geometrie der Zahlen).

Noun

1. Template:Lb The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers.

   Template:Ux

   • 1969, Template:W, 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 $n$-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice.

   • Template:Quote-text
   • 2006, Enrico Bombieri, Walter Gubler, Heights in Diophantine Geometry, Template:W, page 181,

     The easy proof is obtained applying the pigeon-hole principle to

     $\{{\alpha }_1{x}_1+\dots +{\alpha }_n{x}_n(\mathrm{mod}1)|x_i=0,\dots N\}$,

     or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume $2^{n+1}$ given by

     $|X_0+{\alpha }_1{X}_1+\dots {\alpha }_n{X}_n|\le N^{-n},|X_i|\le N,i=1,\dots N$.

Translations

• German: Template:T
• Italian: Template:T

Further reading

• Template:Pedia