Minkowski hypothesis

on the product of inhomogeneous linear forms

A statement according to which for real linear forms

$$ L _ {j} ( \overline{x} ) = a _ {j1} x _ {1} + \dots + a _ {jn} x _ {n} ,\ \quad 1 \leq j \leq n, $$

in $ n $ variables $ x _ {1}, \ldots, x _ {n} $, with a non-zero determinant $ \Delta $, and any reals $ \alpha _ {1}, \ldots ,\alpha _ {n} $, there are integers $ x _ {1}, \ldots, x _ {n} $ such that the inequality

\begin{equation}\label{eq:1} \prod_{j=1}^n | L _ {j} ( \overline{x} ) - \alpha _ {j} | \leq 2^{-n} | \Delta | \end{equation}

holds. This hypothesis was proved by H. Minkowski (1918) in case $ n = 2 $. A proof of the hypothesis is known (1982) for $ n \leq 5 $, and \eqref{eq:1} has been proved for $ n > 5 $ under certain additional restrictions (see [2]).

References

[1]	J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972) Zbl 0209.34401
[2]	B.F. Skubenko, "A proof of Minkowski's conjecture on the product of $n$ linear inhomogeneous forms in $n$ variables for $n \leq 5$" , 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)