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The problem about the homogeneous arithmetical minima of positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470201.png" />-ary quadratic forms with real coefficients. It is equivalent to the problem of the densest lattice packing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470202.png" />-dimensional balls of equal radius (see [[Geometry of numbers|Geometry of numbers]]).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470204.png" />, be a positive quadratic form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470205.png" /> of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470206.png" />; and let
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<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/h/h047/h047020/h0470207.png" /></td> </tr></table>
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The problem about the homogeneous arithmetical minima of positive  $  n $-
 +
ary quadratic forms with real coefficients. It is equivalent to the problem of the densest lattice packing of  $  n $-
 +
dimensional balls of equal radius (see [[Geometry of numbers|Geometry of numbers]]).
 +
 
 +
Let  $  f = f ( x) $,
 +
$  x \in \mathbf R  ^ {n} $,
 +
be a positive quadratic form over  $  \mathbf R $
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of determinant  $  d = d ( f  ) = \mathop{\rm det}  f \neq 0 $;  
 +
and let
 +
 
 +
$$
 +
m ( f  )  = \inf _ {\begin{array}{c}
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x \in \mathbf Z  ^ {n} \\
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 +
x \neq 0
 +
\end{array}
 +
}
 +
f ( x)  = \min _ {\begin{array}{c}
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x \in \mathbf Z  ^ {n} \\
 +
 
 +
{x \neq 0 }
 +
\end{array}
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}  f ( x)
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$$
  
 
be its homogeneous arithmetical minimum. The quantity
 
be its homogeneous arithmetical minimum. The quantity
  
<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/h/h047/h047020/h0470208.png" /></td> </tr></table>
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$$
 +
\gamma _ {n}  = \sup 
 +
\frac{m ( f  ) }{\{ d ( f  ) \}  ^ {1/n} }
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  = \max \
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 +
\frac{m ( f  ) }{\{ d ( f  ) \}  ^ {1/n} }
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,
 +
$$
  
where the supermum or maximum is over all positive quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h0470209.png" />, is called the Hermite constant; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702011.png" /> is the radial function corresponding to a ball.
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where the supermum or maximum is over all positive quadratic forms $  f $,  
 +
is called the Hermite constant; $  \gamma _ {n} = \{ \gamma ( F _ {n} ) \}  ^ {2} $,  
 +
where $  F _ {n} ( x) = ( x _ {1}  ^ {2} + \dots + x _ {n}  ^ {2} )  ^ {1/2} $
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is the radial function corresponding to a ball.
  
Originally, one understood by the Hermite problem the task of finding or estimating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702012.png" /> (from above and below). The exact values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702013.png" /> are known only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702014.png" /> (see [[#References|[1]]]). For estimates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702015.png" />, see [[#References|[2]]] or [[#References|[1]]].
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Originally, one understood by the Hermite problem the task of finding or estimating $  \gamma _ {n} $(
 +
from above and below). The exact values of $  \gamma _ {n} $
 +
are known only for $  n \leq  8 $(
 +
see [[#References|[1]]]). For estimates of $  \gamma _ {n} $,  
 +
see [[#References|[2]]] or [[#References|[1]]].
  
Subsequently, the term Hermite problem was used for the search for local maxima (boundary or extremal) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702016.png" /> in the space of coefficients and their corresponding forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047020/h04702017.png" />. Algorithms are known for finding all classes of boundary forms. In particular, the Voronoi algorithm for perfect forms (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]).
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Subsequently, the term Hermite problem was used for the search for local maxima (boundary or extremal) of $  m ( f  ) / \{ d ( f  ) \}  ^ {1/n} $
 +
in the space of coefficients and their corresponding forms $  f $.  
 +
Algorithms are known for finding all classes of boundary forms. In particular, the Voronoi algorithm for perfect forms (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]).
  
 
The problem was posed by Ch. Hermite in 1850.
 
The problem was posed by Ch. Hermite in 1850.

Latest revision as of 22:10, 5 June 2020


The problem about the homogeneous arithmetical minima of positive $ n $- ary quadratic forms with real coefficients. It is equivalent to the problem of the densest lattice packing of $ n $- dimensional balls of equal radius (see Geometry of numbers).

Let $ f = f ( x) $, $ x \in \mathbf R ^ {n} $, be a positive quadratic form over $ \mathbf R $ of determinant $ d = d ( f ) = \mathop{\rm det} f \neq 0 $; and let

$$ m ( f ) = \inf _ {\begin{array}{c} x \in \mathbf Z ^ {n} \\ x \neq 0 \end{array} } f ( x) = \min _ {\begin{array}{c} x \in \mathbf Z ^ {n} \\ {x \neq 0 } \end{array} } f ( x) $$

be its homogeneous arithmetical minimum. The quantity

$$ \gamma _ {n} = \sup \frac{m ( f ) }{\{ d ( f ) \} ^ {1/n} } = \max \ \frac{m ( f ) }{\{ d ( f ) \} ^ {1/n} } , $$

where the supermum or maximum is over all positive quadratic forms $ f $, is called the Hermite constant; $ \gamma _ {n} = \{ \gamma ( F _ {n} ) \} ^ {2} $, where $ F _ {n} ( x) = ( x _ {1} ^ {2} + \dots + x _ {n} ^ {2} ) ^ {1/2} $ is the radial function corresponding to a ball.

Originally, one understood by the Hermite problem the task of finding or estimating $ \gamma _ {n} $( from above and below). The exact values of $ \gamma _ {n} $ are known only for $ n \leq 8 $( see [1]). For estimates of $ \gamma _ {n} $, see [2] or [1].

Subsequently, the term Hermite problem was used for the search for local maxima (boundary or extremal) of $ m ( f ) / \{ d ( f ) \} ^ {1/n} $ in the space of coefficients and their corresponding forms $ f $. Algorithms are known for finding all classes of boundary forms. In particular, the Voronoi algorithm for perfect forms (see [1], [3], [4]).

The problem was posed by Ch. Hermite in 1850.

References

[1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) MR0893813 Zbl 0611.10017
[2] C.A. Rogers, "Packing and covering" , Cambridge Univ. Press (1964) MR0172183 Zbl 0176.51401
[3] B.N. Delone, "The Peterburg school of number theory" , Moscow-Leningrad (1947) (In Russian)
[4] E.P. Baranovskii, "Packings, coverings, partitionings and certain other distributions in spaces of constant curvature" Progress in Math. , 9 (1971) pp. 209–253 Itogi Nauk. Mat. Algebra Topol. Geom. 1967 (1969) pp. 189–225
How to Cite This Entry:
Hermite problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_problem&oldid=24079
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article