Difference between revisions of "Hermite problem"
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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 $ | ||
| + | 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 | 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 | + | 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 | + | 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 | + | 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 |
Hermite problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_problem&oldid=24079