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 formula_1 and the study of these lattices provides fundamental information on algebraic numbers. Hermann Minkowski (1896) initiated this line of research at the age of 26 in his work "The Geometry of Numbers".
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 results.
Suppose that formula_2 is a lattice in formula_3-dimensional Euclidean space formula_4 and formula_5 is a convex centrally symmetric body.
Minkowski's theorem, sometimes called Minkowski's first theorem, states that if formula_6, then formula_5 contains a nonzero vector in formula_2.
The successive minimum formula_9 is defined to be the inf of the numbers formula_10 such that formula_11 contains formula_12 linearly independent vectors of formula_2.
Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that
formula_14
Later research in the geometry of numbers.
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 of W. M. Schmidt.
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 "L"1...,"L""n" 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
formula_15
lie in a finite number of proper subspaces of Q"n".
Influence on functional analysis.
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. Minkowski's theorem was generalized to topological vector spaces by Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.
Researchers continue to study generalizations to star-shaped sets and other non-convex sets.
References.
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