What is the probability that a random integral quadratic form in n variables has an integral zero?
M. Bhargava, J. E. Cremona, T. A. Fisher, N. G. Jones, J. P. Keating
Abstract
We show that the density of quadratic forms in
n
variables over
Z
p
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
n
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
n
, we determine an exact expression for the probability that a random integral quadratic form in
n
variables is isotropic (i.e., has a nontrivial zero over
Z
), 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 has an integral zero; numerically, this probability is approximately
98.3%
.