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A new arithmetic invariant E(f) is defined for integral binary quadratic forms f. It has the property that, denoting by fm the norm-form of a quadratic number field Q(m), E(fm) < 1 if and only if Q(m) has class number one.

A convex polytope D (α) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients α1, α2,…αn. A local minimum of the determinant D(f) over D(α) must occur at a vertex of D(α). Here a criterion is obtained for a given vertex to provide a local minimum, completely analogous to Voronoï's criterion for a perfect form to be extreme.

A fundamental region for the reduction of positive quaternary quadratic forms is exhibited. It is a convex polyhedral cone with twelve edges in the 10-dimensional space of quaternary quadratic forms.

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).

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.

Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.

All extreme lattice-coverings ofn–space by spheres are known forn≦ 4; see for example [3]. Only one class of extreme covering is known for largen, namely that corresponding to the quadratic formthis was first shown to be extreme for alln≧ 2 by Bleicher [2].

It is well known that the problem of determining the most economical covering ofn-dimensional Eucidean space, by equal spheres whose centres form a lattice, may be formulated in terms of positive definite quadratic forms, as follows: Letbe positive definite, andFor real α, set(the minimum being taken over integralx)

A positive quadratic form, of determinantand minimum M for integral, is said to be extreme if the ratiois a (local) maximum for small variations in the coefficients.Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.