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$$  
 
$$  
L _ {j} ( \overline{x}\; )  = \
+
L _ {j} ( \overline{x} )  = a _ {j1} x _ {1} + \dots + a _ {jn} x _ {n} ,\ \quad 1 \leq  j \leq  n,
a _ {j1} x _ {1} + \dots + a _ {jn} x _ {n} ,\ \  
 
1 \leq  j \leq  n,
 
 
$$
 
$$
  
 
in  $  n $
 
in  $  n $
variables  $  x _ {1} \dots x _ {n} $,  
+
variables  $  x _ {1}, \ldots, x _ {n} $,  
 
with a non-zero determinant  $  \Delta $,  
 
with a non-zero determinant  $  \Delta $,  
and any real $  \alpha _ {1} \dots \alpha _ {n} $,  
+
and any reals $  \alpha _ {1}, \ldots ,\alpha _ {n} $,  
there are integers  $  x _ {1} \dots x _ {n} $
+
there are integers  $  x _ {1}, \ldots, x _ {n} $
 
such that the inequality
 
such that the inequality
  
$$ \tag{* }
+
\begin{equation}\label{eq:1}
\prod _ { j= } 1 ^ { n }
+
\prod_{j=1}^n | L _ {j} ( \overline{x} ) - \alpha _ {j} | \leq 2^{-n} | \Delta |
| L _ {j} ( \overline{x}\; ) - \alpha _ {j} |
+
\end{equation}
\leq   2 ^ {-} n | \Delta |
 
$$
 
  
 
holds. This hypothesis was proved by H. Minkowski (1918) in case  $  n = 2 $.  
 
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 $,  
 
A proof of the hypothesis is known (1982) for  $  n \leq  5 $,  
and (*) has been proved for  $  n > 5 $
+
and \eqref{eq:1} has been proved for  $  n > 5 $
 
under certain additional restrictions (see [[#References|[2]]]).
 
under certain additional restrictions (see [[#References|[2]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,   "An introduction to the geometry of numbers" , Springer  (1972)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.F. Skubenko,   "A proof of Minkowski's conjecture on the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405011.png" /> linear inhomogeneous forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405012.png" /> variables for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405013.png" />" , ''Investigations in number theory'' , ''Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov'' , '''33'''  (1973)  pp. 6–36  (In Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer  (1972) {{ZBL|0209.34401}}</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====

Latest revision as of 20:27, 11 November 2023


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)
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