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Equidistribution of values of linear forms on a cubic hypersurface

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Chow, Sam (2017) Equidistribution of values of linear forms on a cubic hypersurface. Algebra & Number Theory, 10 (2). pp. 421-450. doi:10.2140/ant.2016.10.421

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Official URL: https://doi.org/10.2140/ant.2016.10.421

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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∈Rr∖{0}, then α⋅L is not a rational form. Assume that h>16+8r. Let τ∈Rr, 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 Rr, requiring only that h>16.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Diophantine equations , Diophantine approximation, Hypersurfaces
Journal or Publication Title: Algebra & Number Theory
Publisher: Mathematical Sciences Publishers
ISSN: 1937-0652
Official Date: 16 November 2017
Dates:
DateEvent
16 November 2017Published
27 December 2015Accepted
Date of first compliant deposit: 16 September 2019
Volume: 10
Number: 2
Page Range: pp. 421-450
DOI: 10.2140/ant.2016.10.421
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
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