Geometry of numbers is the part of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
which uses
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
for the study of
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s. Typically, a
ring of algebraic integers
In algebraic number theory, an algebraic integer is a complex number which is Integral element, integral over the Integer#Algebraic properties, integers. That is, an algebraic integer is a complex root of a polynomial, root of some monic polyno ...
is viewed as a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
and
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated by r ...
, the problem of finding
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s that approximate an
irrational quantity.
Minkowski's results
Suppose that
is a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
in
-dimensional Euclidean space
and
is a convex centrally symmetric body.
Minkowski's theorem
In mathematics, Minkowski's theorem is the statement that every convex set in \mathbb^n which is symmetric with respect to the origin and which has volume greater than 2^n contains a non-zero integer point (meaning a point in \Z^n that is not t ...
, sometimes called Minkowski's first theorem, states that if
, then
contains a nonzero vector in
.
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
:
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
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educati ...
,
Harold Davenport
Harold Davenport FRS (30 October 1907 – 9 June 1969) was an English mathematician, known for his extensive work in number theory.
Early life
Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar Scho ...
and
Carl Ludwig Siegel
Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
). 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
In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by .
Statement
The subspace theorem states that if ''L''1,...,''L'n'' are linearly independ ...
was obtained by
Wolfgang M. Schmidt
Wolfgang M. Schmidt (born 3 October 1933) is an Austrian mathematician working in the area of number theory. He studied mathematics at the University of Vienna, where he received his PhD, which was supervised by Edmund Hlawka, in 1955. Wolfgang ...
in 1972. It states that if ''n'' is a positive integer, and ''L''
1,...,''L''
''n'' are
linearly independent
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
forms
Form is the shape, visual appearance, or configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to write or enter data
* ...
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 Q
''n''.
Influence on functional analysis
Minkowski's geometry of numbers had a profound influence on
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. Minkowski proved that symmetric convex bodies induce
norms in finite-dimensional vector spaces. Minkowski's theorem was generalized to
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s by
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, whose theorem states that the symmetric convex sets that are closed and bounded generate the topology of a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
Researchers continue to study generalizations to
star-shaped set
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...
s and other
non-convex sets.
[Kalton et alii. Gardner]
References
Bibliography
* Matthias Beck, Sinai Robins. ''
Computing the continuous discretely: Integer-point enumeration in polyhedra'',
Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics (UTM) (ISSN 0172-6056) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow boo ...
, Springer, 2007.
*
*
*
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
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
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
* (Republished in 1964 by Dover.)
*
Edmund Hlawka
Edmund Hlawka (November 5, 1916, Bruck an der Mur, Styria – February 19, 2009) was an Austrian mathematician. He was a leading number theorist. Hlawka did most of his work at the Vienna University of Technology. He was also a visiting profes ...
, Johannes Schoißengeier, Rudolf Taschner. ''Geometric and Analytic Number Theory''. Universitext. Springer-Verlag, 1991.
*
*
C. G. Lekkerkererker. ''Geometry of Numbers''. Wolters-Noordhoff, North Holland, Wiley. 1969.
*
*
Lovász, L.: ''An Algorithmic Theory of Numbers, Graphs, and Convexity'', CBMS-NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania, 1986
*
*
*
Wolfgang M. Schmidt
Wolfgang M. Schmidt (born 3 October 1933) is an Austrian mathematician working in the area of number theory. He studied mathematics at the University of Vienna, where he received his PhD, which was supervised by Edmund Hlawka, in 1955. Wolfgang ...
. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980
996 with minor corrections
*
*
* 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
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
. Theory of reduction for arithmetical equivalence . Trans. Amer. Math. Soc. 48 (1940) 126–164.
* Hermann Weyl. Theory of reduction for arithmetical equivalence. II . Trans. Amer. Math. Soc. 51 (1942) 203–231.
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