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''on the product of inhomogeneous linear forms''
 
''on the product of inhomogeneous linear forms''
  
 
A statement according to which for real linear forms
 
A statement according to which for real linear forms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640501.png" /></td> </tr></table>
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$$
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L _ {j} ( \overline{x}\; = \
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a _ {j1} x _ {1} + \dots + a _ {jn} x _ {n} ,\ \
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1 \leq  j \leq  n,
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$$
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640502.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640503.png" />, with a non-zero determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640504.png" />, and any real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640505.png" />, there are integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640506.png" /> such that the inequality
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in $  n $
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variables $  x _ {1} \dots x _ {n} $,  
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with a non-zero determinant $  \Delta $,  
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and any real $  \alpha _ {1} \dots \alpha _ {n} $,  
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there are integers $  x _ {1} \dots x _ {n} $
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such that the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ \tag{* }
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\prod _ { j= } 1 ^ { n }
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| L _ {j} ( \overline{x}\; ) - \alpha _ {j} |
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\leq  2  ^ {-} n | \Delta |
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$$
  
holds. This hypothesis was proved by H. Minkowski (1918) in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640508.png" />. A proof of the hypothesis is known (1982) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m0640509.png" />, and (*) has been proved for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064050/m06405010.png" /> under certain additional restrictions (see [[#References|[2]]]).
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holds. This hypothesis was proved by H. Minkowski (1918) in case $  n = 2 $.  
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A proof of the hypothesis is known (1982) for $  n \leq  5 $,  
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and (*) has been proved for $  n > 5 $
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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)</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>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:00, 6 June 2020


on the product of inhomogeneous linear forms

A statement according to which for real linear forms

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

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

$$ \tag{* } \prod _ { j= } 1 ^ { n } | L _ {j} ( \overline{x}\; ) - \alpha _ {j} | \leq 2 ^ {-} n | \Delta | $$

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 (*) 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)
[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=12742
This article was adapted from an original article by E.I. Kovalevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article