What is the probability that a random integral quadratic form in $n$ variables is isotropic?

TL;DR

This paper calculates the probability that a random integral quadratic form in n variables is isotropic, showing it is a rational function in p independent of p, with explicit formulas and specific probabilities for quaternary forms.

Contribution

It explicitly determines the rational function describing the density of isotropic quadratic forms over Z_p and computes the asymptotic probabilities for random integral forms.

Findings

Probability for quaternary forms is approximately 97.0% with uniform coefficients.

Probability increases to approximately 98.3% under GOE distribution.

The density function is a p-independent rational function explicitly determined.

Abstract

We show that the density of quadratic forms in $n$ variables over ${\mathbb Z}_p$ that are isotropic is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a consequence, for each $n$, we determine the probability that a random integral quadratic form in $n$ variables is isotropic. In particular, we show that the probability that a random integral quaternary quadratic form is isotropic is $\approx 97.0\%$, in the case where the coefficients of the quadratic form are independently and uniformly distributed in the range $[-X,X]$ with $X\to\infty$. When random integral quaternary quadratic forms are chosen with respect to the Gaussian Orthogonal Ensemble (GOE), the probability of isotropy increases to $\approx 98.3\%$.