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

We show that the density of quadratic forms in $n$ variables over $\mathbb {Z}_p$ 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 $n$ 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%. [ABSTRACT FROM AUTHOR]

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