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Rydin Myerson, Simon L. (2019) Systems of cubic forms in many variables. Journal fur die reine und angewandte Mathematik, 2019 (757). pp. 309-328. doi:10.1515/crelle-2017-0040 ISSN 0075-4102.
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WRAP-systems-cubic-forms-many-variables-Myerson-2019.pdf - Published Version - Requires a PDF viewer. Download (978Kb) | Preview |
Official URL: https://doi.org/10.1515/crelle-2017-0040
Abstract
We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided n≥25R, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in 25R variables, previous work having required that n≫R2. One conjectures that n≥6R+1 should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Equations, Cubic , Variables (Mathematics), Forms (Mathematics) | |||||||||
| Journal or Publication Title: | Journal fur die reine und angewandte Mathematik | |||||||||
| Publisher: | Walter de Gruyter & Co | |||||||||
| ISSN: | 0075-4102 | |||||||||
| Official Date: | 2019 | |||||||||
| Dates: |
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| Volume: | 2019 | |||||||||
| Number: | 757 | |||||||||
| Page Range: | pp. 309-328 | |||||||||
| DOI: | 10.1515/crelle-2017-0040 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Date of first compliant deposit: | 21 December 2020 | |||||||||
| Date of first compliant Open Access: | 22 December 2020 | |||||||||
| RIOXX Funder/Project Grant: |
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