View source for Hermite problem

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]]).

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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be its homogeneous arithmetical minimum. The quantity

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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.

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]]].

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]]]).

The problem was posed by Ch. Hermite in 1850.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) (Updated reprint) {{MR|0893813}} {{ZBL|0611.10017}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C.A. Rogers, "Packing and covering" , Cambridge Univ. Press (1964) {{MR|0172183}} {{ZBL|0176.51401}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.N. Delone, "The Peterburg school of number theory" , Moscow-Leningrad (1947) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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</TD></TR></table>