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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
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: |
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| 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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