What is the probability that a random integral quadratic form in n variables is isotropic?

2013/11/21 by Manjul Bhargava, Bhargava, Manjul, J. E. Cremona +3
Mathematics · #11D09 #11S05 #60B20 #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Stochastic processes and statistical mechanics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1311.5543

openalex publication_date 2013/11/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We show that the density of quadratic forms in n variables over \mathbb Z_p that are isotropic is a rational function in p, where the rational function is independent of p, and we determine this rational function explicitly. As a consequence, for each n, we determine the probability that a random integral quadratic form in n variables is isotropic. In particular, we show that the probability that a random integral quaternary quadratic form is isotropic is ≈ 97.0%, in the case where the coefficients of the quadratic form are independently and uniformly distributed in the range [-X,X] with X→∞. When random integral quaternary quadratic forms are chosen with respect to the Gaussian Orthogonal Ensemble (GOE), the probability of isotropy increases to ≈ 98.3%.

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