In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space. The geometry of numbers was initiated by Hermann Minkowski (1910).
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.
Suppose that Γ is a lattice in n-dimensional Euclidean space Rn and K is a convex centrally symmetric body.
Minkowski's theorem, sometimes called Minkowski's first theorem, states that if , then K contains a nonzero vector in Γ.
The successive minimum λk is defined to be the inf of the numbers λ such that λK contains k linearly independent vectors of Γ.
Minkowski's theorem on successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that
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 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.
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.
- ^ MSC classification, 2010, available at [www.ams.org], Classification 11HXX.
- ^ Schmidt's books. Grötschel et alii, Lovász et alii, Lovász.
- ^ Cassels (1971) p. 203
- ^ Grötschel et alii, Lovász et alii, Lovász, and Beck and Robins.
- ^ Schmidt, Wolfgang M. Norm form equations. Ann. Math. (2) 96 (1972), pp. 526-551.
See also Schmidt's books; compare Bombieri and Vaaler and also Bombieri and Gubler.
- ^ For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et alii.
- ^ Kalton et alii. Gardner
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- Enrico Bombieri and Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.
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- C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.
- Lenstra, A. K.; Lenstra, H. W., Jr.; Lovász, L. (1982). "Factoring polynomials with rational coefficients". Mathematische Annalen. 261 (4): 515–534. doi:10.1007/BF01457454. hdl:1887/3810. MR 0682664.
- Lovász, L.: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
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- Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
- Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. 1467 (2nd ed.). Springer-Verlag. ISBN 3-540-54058-X. Zbl 0754.11020.
- Siegel, Carl Ludwig (1989). Lectures on the Geometry of Numbers. Springer-Verlag.
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