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We show that the density of quadratic forms in   $n$   variables over   $\mathbb {Z}_p$   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   $\mathbb {Z}$  ), 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 is isotropic; numerically, this probability of isotropy is approximately 98.3%.

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