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What is the probability that a random integral quadratic form in n variables has an integral zero?
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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. 38283848. doi:10.1093/imrn/rnv251 ISSN 10737928.

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Official URL: http://dx.doi.org/10.1093/imrn/rnv251
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:  10737928  
Official Date:  1 January 2016  
Dates: 


Volume:  2016  
Number:  12  
Page Range:  pp. 38283848  
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:  MS1001828 SF NSF, EP/K034383/1 EPSRC, FA86551013088 AFOSR 
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