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- W2950847977 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 $98.3%$." @default.
- W2950847977 created "2019-06-27" @default.
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- W2950847977 date "2015-02-20" @default.
- W2950847977 modified "2023-09-28" @default.
- W2950847977 title "What is the probability that a random integral quadratic form in $n$ variables has an integral zero?" @default.
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