Bhargava, Manjul, Cremona, J. E., Fisher, Tom, Jones, Nick G. and Keating, Jonathan P. (2016) What is the probability that a random integral quadratic form in n variables has an integral zero? International Mathematics Research Notices, 2016 (12). pp. 3828-3848. doi:10.1093/imrn/rnv251 ISSN 1073-7928.

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Abstract

We show that the density of quadratic forms in nn variables over ZpZp that are isotropic is a rational function of pp, where the rational function is independent of pp, and we determine this rational function explicitly. When real quadratic forms in nn 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 nn, we determine an exact expression for the probability that a random integral quadratic form in nn variables is isotropic (i.e., has a nontrivial zero over ZZ), 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%.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Forms, Quadratic, Gaussian measures
Journal or Publication Title:	International Mathematics Research Notices
Publisher:	Oxford University Press
ISSN:	1073-7928
Official Date:	1 January 2016
Dates:	Date Event 1 January 2016 Published 9 September 2015 Available 27 July 2015 Accepted 20 March 2015 Submitted
Volume:	2016
Number:	12
Page Range:	pp. 3828-3848
DOI:	10.1093/imrn/rnv251
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Date of first compliant deposit:	14 January 2016
Date of first compliant Open Access:	19 February 2016
Funder:	Simons Foundation (SF), National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), Leverhulme Trust (LT), Royal Society (Great Britain). Wolfson Research Merit Award (RSWRMA), Royal Society (Great Britain). Leverhulme Senior Research Fellowship (RSLSRF), United States. Air Force. Office of Scientific Research (AFOSR)
Grant number:	MS-1001828 SF NSF, EP/K034383/1 EPSRC, FA8655-10-1-3088 AFOSR

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