View source for Minkowski hypothesis

''on the product of inhomogeneous 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>

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

<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>

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]]]).

====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>



====Comments====
See also [[Geometry of numbers|Geometry of numbers]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,   C.G. Lekkerkerker,   "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös,   P.M. Gruber,   J. Hammer,   "Lattice points" , Longman  (1989)</TD></TR></table>