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Minkowski hypothesis

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on the product of inhomogeneous linear forms

A statement according to which for real linear forms

in variables , with a non-zero determinant , and any real , there are integers such that the inequality

(*)

holds. This hypothesis was proved by H. Minkowski (1918) in case . A proof of the hypothesis is known (1982) for , and (*) has been proved for under certain additional restrictions (see [2]).

References

[1] J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972)
[2] B.F. Skubenko, "A proof of Minkowski's conjecture on the product of linear inhomogeneous forms in variables for " , Investigations in number theory , Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov , 33 (1973) pp. 6–36 (In Russian)


Comments

See also Geometry of numbers.

References

[a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2] P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989)
How to Cite This Entry:
Minkowski hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minkowski_hypothesis&oldid=47852
This article was adapted from an original article by E.I. Kovalevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article