# American Institute of Mathematical Sciences

2018, 12: 9-16. doi: 10.3934/jmd.2018002

## Values of random polynomials at integer points

 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 12, 2017 Revised  November 10, 2017 Published  February 2018

Fund Project: 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

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.

Citation: Jayadev S. Athreya, Gregory A. Margulis. Values of random polynomials at integer points. Journal of Modern Dynamics, 2018, 12: 9-16. doi: 10.3934/jmd.2018002
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##### References:
 [1] Alex Eskin, Gregory Margulis and Shahar Mozes. On a quantitative version of the Oppenheim conjecture. Electronic Research Announcements, 1995, 1: 124-130. [2] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [3] Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267 [4] Uri Shapira. On a generalization of Littlewood's conjecture. Journal of Modern Dynamics, 2009, 3 (3) : 457-477. doi: 10.3934/jmd.2009.3.457 [5] Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006 [6] Yakov Varshavsky. A proof of a generalization of Deligne's conjecture. Electronic Research Announcements, 2005, 11: 78-88. [7] Francisco Brito, Maria Luiza Leite, Vicente de Souza Neto. Liouville's formula under the viewpoint of minimal surfaces. Communications on Pure & Applied Analysis, 2004, 3 (1) : 41-51. doi: 10.3934/cpaa.2004.3.41 [8] Marius Mitrea. On Bojarski's index formula for nonsmooth interfaces. Electronic Research Announcements, 1999, 5: 40-46. [9] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421 [10] Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195 [11] Laurent Desvillettes, Clément Mouhot, Cédric Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. Kinetic & Related Models, 2011, 4 (1) : 277-294. doi: 10.3934/krm.2011.4.277 [12] Adriano Regis Rodrigues, César Castilho, Jair Koiller. Vortex pairs on a triaxial ellipsoid and Kimura's conjecture. Journal of Geometric Mechanics, 2018, 10 (2) : 189-208. doi: 10.3934/jgm.2018007 [13] Changfeng Gui. On some problems related to de Giorgi’s conjecture. Communications on Pure & Applied Analysis, 2003, 2 (1) : 101-106. doi: 10.3934/cpaa.2003.2.101 [14] Xijun Hu, Penghui Wang. Hill-type formula and Krein-type trace formula for $S$-periodic solutions in ODEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 763-784. doi: 10.3934/dcds.2016.36.763 [15] Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533 [16] Yong Li, Hongren Wang, Xue Yang. Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2607-2623. doi: 10.3934/dcdsb.2018123 [17] Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic & Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479 [18] Richard Moeckel. A proof of Saari's conjecture for the three-body problem in $\R^d$. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 631-646. doi: 10.3934/dcdss.2008.1.631 [19] Jan Hladký, Diana Piguet, Miklós Simonovits, Maya Stein, Endre Szemerédi. The approximate Loebl-Komlós-Sós conjecture and embedding trees in sparse graphs. Electronic Research Announcements, 2015, 22: 1-11. doi: 10.3934/era.2015.22.1 [20] Wafa Hamrouni, Ali Abdennadher. Random walk's models for fractional diffusion equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2509-2530. doi: 10.3934/dcdsb.2016058

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