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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. (2015) What is the probability that a random integral quadratic form in n variables has an integral zero? International Mathematics Research Notices . rnv251. doi:10.1093/imrn/rnv251
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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 | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Forms, Quadratic | ||||||||
| Journal or Publication Title: | International Mathematics Research Notices | ||||||||
| Publisher: | Oxford University Press | ||||||||
| ISSN: | 1073-7928 | ||||||||
| Official Date: | 9 September 2015 | ||||||||
| Dates: |
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| Number of Pages: | 21 | ||||||||
| Article Number: | rnv251 | ||||||||
| DOI: | 10.1093/imrn/rnv251 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Funder: | 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), United States. Air Force (USAF), University of Warwick, Research Councils UK (RCUK) | ||||||||
| Grant number: | DMS- 1001828 (NSF), EP/K034383/1 (EPSRC), FA8655-10-1-3088 (USAF) |
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