TY - JOUR
T1 - What is the probability that a random integral quadratic form in nvariables has an integral zero?
AU - Bhargava, Manjul
AU - Cremona, John E.
AU - Fisher, Tom
AU - Jones, Nick G.
AU - Keating, Jonathan P.
N1 - Publisher Copyright:
© The Author(s) 2015.
PY - 2016
Y1 - 2016
N2 - 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%.
AB - 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%.
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U2 - 10.1093/imrn/rnv251
DO - 10.1093/imrn/rnv251
M3 - Article
AN - SCOPUS:84981308718
SN - 1073-7928
VL - 2016
SP - 3828
EP - 3848
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 12
ER -