geometry of numbers

English

Etymology

The field was initiated by German mathematician Hermann Minkowski (1910, Geometrie der Zahlen).

Noun

geometry of numbers (uncountable)

1. (number theory) The subbranch of number theory which applies techniques from geometry to the study of algebraic numbers.

   A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in $\textstyle \mathbb {R} ^{n}$; the study of these lattices provides fundamental information on 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 $n$-dimensional euclidean space. Its aim is to study various quantities describing the behaviour of a body with respect to a lattice.

   • 2000, C. D. Olds, Anneli Lax, Giuliana Davidoff, The Geometry of Numbers‎^[1], Mathematical Association of America:
   • 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

     $\{\alpha _{1}x_{1}+\dots +\alpha _{n}x_{n}{\pmod {1}}\vert x_{i}=0,\dots N\}$,

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

     $\vert X_{0}+\alpha _{1}X_{1}+\dots \alpha _{n}X_{n}\vert \leq N^{-n},\vert X_{i}\vert \leq N,i=1,\dots N$.

Translations

subbranch of number theory

• German: Geometrie der Zahlen f
• Italian: teoria dei numeri geometrica f

Further reading

• Category:Geometry of numbers on Wikipedia.Wikipedia