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Quadratic forms and systems of forms in many variables
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Rydin Myerson, Simon L. (2018) Quadratic forms and systems of forms in many variables. Inventiones Mathematicae, 213 (1). pp. 205-235. doi:10.1007/s00222-018-0789-x ISSN 0020-9910.
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RydinMyerson2018_Article_QuadraticFormsAndSystemsOfForm.pdf - Published Version - Requires a PDF viewer. Available under License Creative Commons Attribution 4.0. Download (662Kb) | Preview |
Official URL: http://dx.doi.org/10.1007/s00222-018-0789-x
Abstract
Let F_1,\ldots ,F_R be quadratic forms with integer coefficients in n variables. When n\ge 9R and the variety V(F_1,\ldots ,F_R) is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for V(F_1,\ldots ,F_R). Previous work in this direction required n to grow at least quadratically with R. We give a similar result for R forms of degree d, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as n> d2^dR. In the case that d\ge 3, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.
Item Type: | Journal Article | ||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Journal or Publication Title: | Inventiones Mathematicae | ||||||||
Publisher: | Springer | ||||||||
ISSN: | 0020-9910 | ||||||||
Official Date: | July 2018 | ||||||||
Dates: |
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Volume: | 213 | ||||||||
Number: | 1 | ||||||||
Page Range: | pp. 205-235 | ||||||||
DOI: | 10.1007/s00222-018-0789-x | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||||
Date of first compliant deposit: | 16 November 2020 | ||||||||
Date of first compliant Open Access: | 16 November 2020 |
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