Geometry of numbers

Geometry of numbers is the part of

ring of algebraic integers is viewed as a lattice

\mathbb {R} ^{n},

and the study of these lattices provides fundamental information on algebraic numbers.[1] 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.^[2]

Minkowski's results

Suppose that $\Gamma$ is a lattice in $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$ and $K$ is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if $\operatorname {vol} (K)>2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma )$ , then $K$ contains a nonzero vector in $\Gamma$ .

The successive minimum $\lambda _{k}$ is defined to be the

inf

of the numbers

\lambda

such that

\lambda K

contains

k

linearly independent vectors of

\Gamma

. Minkowski's theorem on

successive minima, sometimes called Minkowski's second theorem, is a strengthening of his first theorem and states that^[3]

\lambda _{1}\lambda _{2}\cdots \lambda _{n}\operatorname {vol} (K)\leq 2^{n}\operatorname {vol} (\mathbb {R} ^{n}/\Gamma ).

Later research in the geometry of numbers

In 1930–1960 research on the geometry of numbers was conducted by many

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.^[4]

Subspace theorem of W. M. Schmidt

In the geometry of numbers, the

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

|L_{1}(x)\cdots L_{n}(x)|<|x|^{-\varepsilon }

lie in a finite number of proper subspaces of Qⁿ.

Influence on functional analysis

Minkowski's geometry of numbers had a profound influence on

Kolmogorov, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a Banach space.^[6]

Researchers continue to study generalizations to

star-shaped sets and other non-convex sets.^[7]

References

^ MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX.
MR 1261419

^ Cassels (1971) p. 203

^ Grötschel et al., Lovász et al., 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 al.

^ Kalton et al. Gardner

Bibliography

Matthias Beck, Sinai Robins. Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, 2007.

S2CID 121274024
.

Enrico Bombieri & Walter Gubler (2006). Heights in Diophantine Geometry. Cambridge U. P.

J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer Classics in Mathematics, Springer-Verlag 1997 (reprint of 1959 and 1971 Springer-Verlag editions).

John Horton Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd ed., 1998.

R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995. Second edition: 2006.

P. M. Gruber, Convex and discrete geometry, Springer-Verlag, New York, 2007.

P. M. Gruber, J. M. Wills (editors), Handbook of convex geometry. Vol. A. B, North-Holland, Amsterdam, 1993.

M. Grötschel, Lovász, L., A. Schrijver: Geometric Algorithms and Combinatorial Optimization, Springer, 1988

Hancock, Harris (1939). Development of the Minkowski Geometry of Numbers. Macmillan. (Republished in 1964 by Dover.)

Edmund Hlawka, Johannes Schoißengeier, Rudolf Taschner. Geometric and Analytic Number Theory. Universitext. Springer-Verlag, 1991.

MR 0808777

C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.

S2CID 5701340
.

Lovász, L.: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986

Malyshev, A.V. (2001) [1994], "Geometry of numbers", Encyclopedia of Mathematics, EMS Press

MR 0249269
, retrieved 2016-02-28

Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])

Zbl 0754.11020
.

Springer-Verlag
.

Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.

Anthony C. Thompson, Minkowski geometry, Cambridge University Press, Cambridge, 1996.

doi:10.1090/S0002-9947-1940-0002345-2

Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231.
doi:10.2307/1989946

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Number theory
Fields

Iwasawa–Tate theory, Kummer theory
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Key concepts

Numbers

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Natural numbers

Unity

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Retrieved from "https://en.wikipedia.org/w/index.php?title=Geometry_of_numbers&oldid=1201049208"

[1] MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX.

[2] MR 1261419

[3] Cassels (1971) p. 203

[4] Grötschel et al., Lovász et al., Lovász, and Beck and Robins.

[5] 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.

[6] For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et al.

[7] Kalton et al. Gardner

[2]

[3]

[4]

[6]

[7]