Normal forms for real quadratic forms

We investigate the non-diagonal normal forms of a quadratic form on , in particular for n = 3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of . Received: 27 June 2008     

Invariants of formally real fields and quadratic forms

Let F be a formally real field. Connections between the invariants I(F), N(F), ud(F), ũd(F), β_F(i) (i ϵ [1, l(F) - 1]) are studied, using a new invariant C(F). The possibilities for β_F are described when l(F) ≤ 5 or ud(F) ≤ 4, without considering existence.     

Cyclic transformations of n-gons and related quadratic forms

This paper discusses circulants that preserve quadratic forms or that are metric dependent. Several related geometric topics are also discussed.     

Universal octonary diagonal forms over some real quadratic fields

In this paper, we will prove there are infinitely many integers n such that n ²— 1 is square-free and admits universal octonary diagonal quadratic forms. Received: November 2, 1998.     

Theta functions of indefinite quadratic forms over real number fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

     

Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints

A symmetric matrix C is called copositive if the quadratic form x′Cx is nonnegative for all nonnegative values of the variables (x₁,x₂,…,x_n)=x′. A known sufficient condition for a quadratic form x′Qx to be positive unless x=0, subject to the linear inequality constraints Ax?0, is that there should exist a copositive matrix C such that Q?A′CA is positive definite. The main result of this paper establishes the necessity of this condition. For x'Qx to be merely nonnegative subject to Ax ? 0, the situation is less straightforward. The necessity of the existence of a copositive matrix C such that Q?A′CA is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of A, and counterexamples are given to show that, in general, no such matrix may exist, even when Slater's constraint qualification holds. Our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x′Qx subject to Ax?0, in which specific symmetric matrices, constructed by rational operations from A and Q and depending upon a single real parameter v, must be tested for positive definiteness or strict copositivity for large values of v. This technique is illustrated by several examples.     

Real perturbation values and real quadratic forms in a complex vector space

A sequence of real numbers connected to a complex matrix is introduced. It is shown how these real perturbation values can be computed and that they have several properties similar to the singular values. The so-called real pseudospectra and real stability radii can be computed using the real perturbation values. The main result concerns the signature of real quadratic forms in complex vector spaces.     

On quadratic homogeneous quasi-translations

In this paper, we describe the structure of quadratic homogeneous quasi-translations, and we prove that, in dimension 9 and less, such a map is linearly triangularizable, or equivalently a quadratic homogeneous nice derivation is linearly conjugate to a triangular derivation.     

Finiteness of real quadratic fields which admit positive integral diagonal septanary universal forms

In this paper, we will prove if D is large enough, there are no positive integral diagonal septanary universal quadratic forms over . Received: 13 November 1997 / Revised version: 17 November 1998     

Representation of integers by positive-definite quadratic forms in totally real algebraic number fields

The results of Gundlach (K. B. Gundlach, Acta Math.,92, 309–345 (1954)) are extended to forms with an odd number of variables. We obtain for alln4 asymptotic formulae for the number of integral representations of numbers by totally positiven-ary forms in totally real algebraic number fields. The remainder term is of the form . The necessary concepts and results about Hilbert modular forms of half integral weight are developed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 32–46, 1983.     

On quadratic forms

The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Canonical forms of pseudo-orthogonal matrices

Canonical forms are found for all real four-dimensional matrices of the pseudo-orthogonal group which differs from the Lorentz group only in that its metric has twoplus signs and twominus signs.     

Quasi-genera of quadratic forms

Let f be a quadratic form in n variables (n > 1) with nonzero determinant d. A prime p is said to be exceptional with respect to f if every automorph of f with rational elements, determinant ±1 and denominator prime to 2d has a denominator which is a quadratic residue of p. (Throughout, slight modifications must be made if p = 2.) Except for certain binary forms, each exceptional prime induces a splitting of the genus into two quasi-genera. Building on previous results, necessary and sufficient conditions are given that a prime p be exceptional for n = 2 and n = 3 and necessary conditions for n > 3. It is proved that there are no exceptional primes for n > 4 and only possibly in special cases for n = 4. A connection is shown between representations of integers by certain ternary forms and the existence of quasi-genera. Possible connections with spinor genera are mentioned and a few unanswered questions are posed.