Values of random polynomials at integer points

Jayadev S. Athreya; Gregory A. Margulis

doi:10.3934/jmd.2018002

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2018, Volume 12: 9-16. Doi: 10.3934/jmd.2018002 open access

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Values of random polynomials at integer points

Jayadev S. Athreya^1, and
Gregory A. Margulis^2,

1.
Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195, USA
2.
Department of Mathematics, Yale University, Box 208283, New Haven, CT 06520, USA

Received: April 11, 2017

Revised: November 09, 2017

Published: February 2018

This material is based upon work while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2015 semester, supported by the National Science Foundation Grant DMS 0932078 000.
JSA: Partially supported by NSF CAREER grant DMS 1559860, NSF grant DMS 1069153, and grants DMS 1107452,1107263,1107367 "RNMS: GEometric structures And Representation varieties (the GEAR Network)".
GAM: Supported by NSF grant DMS 1265695.

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Abstract

Using classical results of Rogers [12, Theorem 1] bounding the L²-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantitative Oppenheim theorem of Eskin-Margulis-Mozes [6] for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.

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Mathematics Subject Classification: Primary: 11H60, 11P21, 11C08.

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References

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