Abstract
We show that the density of quadratic forms in nn variables over ZpZp that are isotropic is a rational function of pp, where the rational function is independent of pp, and we determine this rational function explicitly. When real quadratic forms in nn variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each nn, we determine an exact expression for the probability that a random integral quadratic form in nn variables is isotropic (i.e., has a nontrivial zero over ZZ), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form is isotropic; numerically, this probability of isotropy is approximately 98.3%.
Highlights
An integral quadratic form Q in n variables is a homogeneous quadratic polynomialQ(x1, x2, . . . , xn) = cij xi xj, (1)1≤i≤ j≤n where all coefficients cij lie in Z
We wish to consider the question: what is the probability that a random integral quadratic form in n variables is isotropic?
We give a complete answer to this question for all n, when integral quadratic forms in n variables are chosen according to the Gaussian Orthogonal
Summary
An integral quadratic form Q in n variables is a homogeneous quadratic polynomial. 1≤i≤ j≤n where all coefficients cij lie in Z. We prove Theorem 1.3, that is, we determine for each n the probability that a random real n-ary quadratic form from the GOE distribution is indefinite We accomplish this by first expressing, as a certain determinantal integral, the probability that an n × n symmetric matrix from the GOE distribution has all positive eigenvalues. We show how this determinantal integral can be evaluated using the de Bruijn identity [4], allowing us to obtain an expression for the probability of positive definiteness in terms of the Pfaffian of an explicit skew-symmetric matrix A, as given in Theorem 1.3 We end this introduction by remarking that the analogues of Theorems 1.2 and 1.4 hold over a general local or global field, respectively. Theorem 1.2 holds over any finite extension of Qp, with the same proof, provided that when making substitutions in the proofs we replace p by a uniformiser, and when computing probabilities we replace p by the order of the residue field
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