Minkowski hypothesis

The Minkowski hypothesis is an assumption according to which for any lattice $L\subset \mathbb {R} ^{n}$ ${\ displaystyle L \ subset \ mathbb {R} ^ {n}}$ ${\ displaystyle L \ subset \ mathbb {R} ^ {n}}$ with qualifier $2^{n}$ ${\ displaystyle 2 ^ {n}}$ $2 ^ {n}$ and any vector $v=(v_{1},v_{2},..,v_{n})$ ${\ displaystyle v = (v_ {1}, v_ {2}, .., v_ {n})}$ ${\ displaystyle v = (v_ {1}, v_ {2}, .., v_ {n})}$ there is an element $x=(x_{1},x_{2},..,x_{n})\in L$ ${\ displaystyle x = (x_ {1}, x_ {2}, .., x_ {n}) \ in L}$ ${\ displaystyle x = (x_ {1}, x_ {2}, .., x_ {n}) \ in L}$ such that

|(x_{1}-v_{1})(x_{2}-v_{2})\dots (x_{n}-v_{n})|\leqslant 1

{\ displaystyle | (x_ {1} -v_ {1}) (x_ {2} -v_ {2}) \ dots (x_ {n} -v_ {n}) | \ leqslant 1}

{\ displaystyle | (x_ {1} -v_ {1}) (x_ {2} -v_ {2}) \ dots (x_ {n} -v_ {n}) | \ leqslant 1}

Happening $n=2$ ${\ displaystyle n = 2}$ $n = 2$ this hypothesis was proved by Minkowski ^[1]
At $n=3$ ${\ displaystyle n = 3}$ $n = 3$ Minkowski’s hypothesis was proved by Remak ^[2]
At $n=4$ ${\ displaystyle n = 4}$ $n = 4$ Minkowski hypothesis proved by Dyson ^[3]
At $n=5$ ${\ displaystyle n = 5}$ $n = 5$ Minkowski hypothesis proved Skubenko ^[4]

Literature

Cassels, J. V. S, Introduction to Number Geometry , trans. from English., M., 1955;

↑ Minkowski, Hermann . Geometrie der Zahlen (Neopr.) . - Leipzig-Berlin: RG Teubner, 1910.
↑ Remak, R., Verallgemeinerung eines Minkowskischen Satzes, I, II. Math. Z., 17 (1923), 1-34; 18 (1924), 173-200.
↑ Dyson, FJ, On the product of four non-homogeneous forms. Ann. of Math. B), 49 A948), 82-109.
↑ Skubenko, BF A new variant of the proof of the inhomogeneous Minkowski conjecture for $ n = 5 $. (Russian) Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 142 (1976), 240--253, 271

[1] Minkowski, Hermann . Geometrie der Zahlen (Neopr.) . - Leipzig-Berlin: RG Teubner, 1910.

[2] Remak, R., Verallgemeinerung eines Minkowskischen Satzes, I, II. Math. Z., 17 (1923), 1-34; 18 (1924), 173-200.

[3] Dyson, FJ, On the product of four non-homogeneous forms. Ann. of Math. B), 49 A948), 82-109.

[4] Skubenko, BF A new variant of the proof of the inhomogeneous Minkowski conjecture for $ n = 5 $. (Russian) Number theory, mathematical analysis and their applications. Trudy Mat. Inst. Steklov. 142 (1976), 240--253, 271

Minkowski hypothesis

Literature

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