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We show that the density of quadratic forms in $n$ variables over ${\BBZ}_p$ 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 probab...

What is the Probability that a Random Integral Quadratic Form innVariables has an Integral Zero?