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• W2114554625 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 is isotropic; numerically, this probability of isotropy is approximately 98.3%." @default.
• W2114554625 created "2016-06-24" @default.
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• W2114554625 date "2015-09-09" @default.
• W2114554625 modified "2023-10-16" @default.
• W2114554625 title "What is the Probability that a Random Integral Quadratic Form in<i>n</i>Variables has an Integral Zero?" @default.
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• W2114554625 doi "https://doi.org/10.1093/imrn/rnv251" @default.
• W2114554625 hasPublicationYear "2015" @default.
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