Hermite problem

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