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What is the probability that a random integral quadratic form in n variables has an integral zero?

Author(s): Bhargava, Manjul; Cremona, JE; Fisher, TA; Jones, NG; Keating, JP

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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 $98.3\%$.
Publication Date: 2016
Electronic Publication Date: 9-Sep-2015
Citation: M. Bhargava, J. E. Cremona, T. Fisher, N. G. Jones, J. P. Keati ng: What is the probability that a random integral quadratic form in n variables has an integral zero ? Int. Math. Res. Not. IMRN, 12 (2016), 3828–3848.
Pages: 3828-3848
Type of Material: Journal Article
Journal/Proceeding Title: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
Version: Author's manuscript



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