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

Manjul Bhargava, John E. Cremona, Tom Fisher, Nick G. Jones, Jonathan P. Keating

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

We show that the density of quadratic forms in nvariables over ℤp 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 nvariables 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 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%.

Original languageEnglish (US)
Pages (from-to)3828-3848
Number of pages21
JournalInternational Mathematics Research Notices
Volume2016
Issue number12
DOIs
StatePublished - 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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