Geometry of numbers

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 $\mathbb R^n,$ and the study of these lattices provides fundamental information on algebraic numbers.MSC classification, 2010, available at http://www.ams.org/msc/msc2010.html, Classification 11HXX. {{harvs|txt|authorlink=Hermann Minkowski|first=Hermann|last= Minkowski|year=1896|ref1=https://mathweb.ucsd.edu/~b3tran/cgm/Minkowski_SpaceAndTime_1909.pdf}} initiated this line of research at the age of 26 in his work The Geometry of Numbers.{{Cite book |last=Minkowski |first=Hermann |url=https://books.google.com/books?id=D-J9AgAAQBAJ&dq=Space+and+Time+Minkowski%E2%80%99s+Papers+on+Relativity&pg=PA1 |title=Space and Time: Minkowski's papers on relativity |date=2013-08-27 |publisher=Minkowski Institute Press |isbn=978-0-9879871-1-2 |language=en}}

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.Schmidt's books. {{Cite Geometric Algorithms and Combinatorial Optimization}}

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 thatCassels (1971) p. 203

: $\lambda_1\lambda_2\cdots\lambda_n \operatorname{vol} (K)\le 2^n \operatorname{vol} (\mathbb{R}^n/\Gamma).$

Later research in the geometry of numbers

In 1930–1960 research on the geometry of numbers was conducted by many 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.Grötschel et al., Lovász et al., Lovász, and Beck and Robins.

=Subspace theorem of W. M. Schmidt=

{{See also|Siegel's lemma|volume (mathematics)|determinant|parallelepiped}}

In the geometry of numbers, the subspace theorem was obtained by Wolfgang M. Schmidt in 1972.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. It states that if n is a positive integer, and L₁,...,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

: $|L_1(x)\cdots L_n(x)|<|x|^{-\varepsilon}$

lie in a finite number of proper subspaces of Qⁿ.

Influence on functional analysis

Minkowski's geometry of numbers had a profound influence on 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.For Kolmogorov's normability theorem, see Walter Rudin's Functional Analysis. For more results, see Schneider, and Thompson and see Kalton et al.

Researchers continue to study generalizations to star-shaped sets and other non-convex sets.Kalton et al. Gardner

References

Bibliography

Matthias Beck, Sinai Robins. Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, 2007.
{{cite journal|author=Enrico Bombieri|author-link=Enrico Bombieri|author2=Vaaler, J.|title = On Siegel's lemma|journal = Inventiones Mathematicae|volume = 73|issue = 1|date = Feb 1983|pages = 11–32|doi = 10.1007/BF01393823|bibcode=1983InMat..73...11B|s2cid=121274024}}
{{cite book

|author=Enrico Bombieri

|author-link=Enrico Bombieri

|author2=Walter Gubler

|name-list-style=amp

|title=Heights in Diophantine Geometry

|publisher=Cambridge U. P.

|year=2006}}

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
{{cite book

| author = Hancock, Harris

| title = Development of the Minkowski Geometry of Numbers

| year = 1939

| publisher = Macmillan}} (Republished in 1964 by Dover.)

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

|last1=Kalton|first1=Nigel J.|author1-link=Nigel Kalton

|last2=Peck|first2=N. Tenney

|last3=Roberts|first3=James W.

| title = An F-space sampler

| series = London Mathematical Society Lecture Note Series, 89

| publisher = Cambridge University Press| location = Cambridge

| year = 1984| pages = xii+240| isbn = 0-521-27585-7 | mr = 0808777}}

C. G. Lekkerkererker. Geometry of Numbers. Wolters-Noordhoff, North Holland, Wiley. 1969.
{{cite journal | author = Lenstra, A. K. | author-link = Arjen Lenstra | author2 = Lenstra, H. W. Jr. | author2-link = Hendrik Lenstra | author3 = Lovász, L. | author3-link = László Lovász | title = Factoring polynomials with rational coefficients | journal = Mathematische Annalen | volume = 261 | year = 1982 | issue = 4 | pages = 515–534 | hdl = 1887/3810 | doi = 10.1007/BF01457454 | mr = 0682664| s2cid = 5701340 | url = http://infoscience.epfl.ch/record/164484/files/nscan4.PDF }}
Lovász, L.: An Algorithmic Theory of Numbers, Graphs, and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
{{Springer|id=G/g044350|title=Geometry of numbers|first=A.V. |last=Malyshev}}
{{Citation | last1=Minkowski | first1=Hermann | author1-link=Hermann Minkowski | title=Geometrie der Zahlen | url=https://archive.org/details/geometriederzahl00minkrich | publisher=R. G. Teubner | location=Leipzig and Berlin | mr=0249269 | year=1910 | jfm=41.0239.03 | access-date=2016-02-28}}
Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
{{cite book | last=Schmidt | first=Wolfgang M. | author-link=Wolfgang M. Schmidt | title=Diophantine approximations and Diophantine equations | series=Lecture Notes in Mathematics | volume=1467 | publisher=Springer-Verlag | year=1996 | edition=2nd | isbn=3-540-54058-X | zbl=0754.11020}}
{{cite book | author = Siegel, Carl Ludwig | author-link = Carl Ludwig Siegel | title = Lectures on the Geometry of Numbers | url = https://archive.org/details/lecturesongeomet0000sieg | url-access = registration | year = 1989 | publisher = 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.
Hermann Weyl. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164. {{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}}