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Concept
Geometry of numbers
Summary
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by .
The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity.
Minkowski's theorem
Suppose that is a lattice in -dimensional Euclidean space and is a convex centrally symmetric body.
Minkowski's theorem, sometimes called Minkowski's first theorem, states that if , then contains a nonzero vector in .
Minkowski's second theorem
The successive minimum is defined to be the inf of the numbers such that contains linearly independent vectors of .
Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that
In 1930-1960 research on the geometry of numbers was conducted by many number theorists (including Louis Mordell, Harold Davenport and Carl Ludwig Siegel). In recent years, Lenstra, Brion, and Barvinok have developed combinatorial theories that enumerate the lattice points in some convex bodies.
Subspace theorem
Siegel's lemmavolume (mathematics)determinant and Parallelepiped
In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972. It states that if n is a positive integer, and L1,...,Ln are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then
the non-zero integer points x in n coordinates with
lie in a finite number of proper subspaces of Qn.
normed vector space
Banach space and F-space
Minkowski's geometry of numbers had a profound influence on functional analysis. Minkowski proved that symmetric convex bodies induce norms in finite-dimensional vector spaces.
Official source
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