Minkowski's fundamental inequality for reduced positive quadratic forms

Abstract

Forms which are reduced in the sense of Minkowski satisfy the “fundamental inequality” a11a22 hellipann≤λnD; the best possible value of λn is known for n≤5. A more precise result for the minimum value of D in terms of the diagonal coefficients has been stated by Oppenheim for ternary forms. The corresponding precise result for quaternary forms is established here by considering a convex polytope D(α), defined as the intersection of the cone of reduced forms with the hyperplanes aii = αi (i = 1, hellip n).

On Gauss's proof of Seeber's Theorem

“…which is best possible for given a, b, c. A proof was published in 1978 by Barnes [1], who also obtained at that time (unpublished) a pair of identities of the Gauss type for (5).…”

“…However, though the polynomial det(a,;) -~a~ i " " " 6t44 is known (cf. [1]) to be positive throughout the reduction domain provided 0 < al~ < a22 < a33 < a44, it is not clear in advance that the coefficients in the polynomial representation will depend rationally on a~ ..... 044. To confirm Minkowski's statement concerning the existence of identities by this method would thus require carrying out the calculations explicitly, according to the method of [5] p. 59, for each of the large number of simpiices to which triangulation of the reduction domain gives rise.…”

“…Here we examine Gauss's method from a general standpoint, as a method whereby, in certain circumstances, a polynomial f(x~ ..... xr) may be shown to be non-negative on a convex polytope P by representing it as a positive multilinear combination of the linear forms which determine P. Among other things, we show AMS (1990) Lyl < a, 1 x +y--z,x --y +z, --x +y +z, --x --y --z ~<~(a +b).…”

A class of D-extreme Minkowski-reduced forms

“…A form in ^D(a) for which the determinant D{f) is a local minimum over all/ in fy(a) is called (Barnes (1978(Barnes ( , 1979 The form f n (x) is absolutely ' TD -extreme for n = 2 and 3 and is a natural generalization of Voronoi's principal perfect form (see Voronoi (1907)…”

Minkowski's fundamental inequality for reduced positive quadratic forms(II)

“…A convex polytope^(ot) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients a u a 2 ,...,«.*. A local minimum of the determinant D(J) over^(a) must occur at a vertex of ^(a).…”