What is the Probability that a Random Integral Quadratic Form in n Variables has an Integral Zero?

Abstract

We show that the density of quadratic forms in n variables over Zp that are isotropic is a rational function of p, where the rational function is independent of p, and we determine this rational function explicitly. When real quadratic forms in 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, we determine an exact expression for the probability that a random integral quadratic form in variables is isotropic (i.e., has a nontrivial zero over), 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%.