Meaning of geometry of numbers | Babel Free

Definitions

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

uncountable

Equivalents

Examples

“A typical approach in the geometry of numbers is to view a ring of algebraic integers as a lattice in #92;textstyle#92;mathbb#123;R#125;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.”

“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 α₁x_1+…+αₙx_n(mod 1)|x_i=0,…N, or 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 |X_0+α₁X_1+…αₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N.”

CEFR level