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

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arXiv · 1502.05992

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

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Abstract

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 has an integral zero; numerically, this probability is approximately $98.3\%$.

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M. Bhargava, J. E. Cremona, T. A. Fisher, N. G. Jones, J. P. Keating. 2015-02-20. What is the probability that a random integral quadratic form in $n$ variables has an integral zero?. https://arxiv.org/abs/1502.05992

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