Rydin Myerson, SL;
(2018)
Quadratic forms and systems of forms in many variables.
Inventiones Mathematicae
, 213
(1)
pp. 205-235.
10.1007/s00222-018-0789-x.
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Abstract
Let F1,…,FRF1,…,FR be quadratic forms with integer coefficients in n variables. When n≥9Rn≥9R and the variety V(F1,…,FR)V(F1,…,FR) 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(F1,…,FR)V(F1,…,FR) . 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>d2dRn>d2dR . In the case that d≥3d≥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.
| Type: | Article |
|---|---|
| Title: | Quadratic forms and systems of forms in many variables |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00222-018-0789-x |
| Publisher version: | https://doi.org/10.1007/s00222-018-0789-x |
| Language: | English |
| Additional information: | © The Author(s) 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
| Keywords: | Forms in many variables, Hardy-Littlewood method, quadratic forms, rational points |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10044411 |
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