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2016 Equidistribution of values of linear forms on a cubic hypersurface
Sam Chow
Algebra Number Theory 10(2): 421-450 (2016). DOI: 10.2140/ant.2016.10.421

Abstract

Let C be a cubic form with integer coefficients in n variables, and let h be the h-invariant of C. Let L1,,Lr be linear forms with real coefficients such that, if α r {0}, then αL is not a rational form. Assume that h > 16 + 8r. Let τ r, and let η be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions x [P,P]n to the system C(x) = 0, |L(x) τ| < η. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the h-invariant condition with the hypothesis n > 16 + 9r and show that the system has an integer solution. Finally, we show that the values of L at integer zeros of C are equidistributed modulo 1 in r, requiring only that h > 16.

Citation

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Sam Chow. "Equidistribution of values of linear forms on a cubic hypersurface." Algebra Number Theory 10 (2) 421 - 450, 2016. https://doi.org/10.2140/ant.2016.10.421

Information

Received: 29 April 2015; Revised: 9 November 2015; Accepted: 27 December 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06561469
MathSciNet: MR3477746
Digital Object Identifier: 10.2140/ant.2016.10.421

Subjects:
Primary: 11D25
Secondary: 11D75 , 11J13 , 11J71 , 11P55

Keywords: diophantine approximation , Diophantine equations , diophantine inequalities , equidistribution

Rights: Copyright © 2016 Mathematical Sciences Publishers

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