Improvements in Birch's theorem on forms in many variables

Browning, Timothy D; Prendiville, Sean

Earlier Version

Improvements in Birch's theorem on forms in many variables

Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many variables. Journal fur die Reine und Angewandte Mathematik. 2017(731), 203–234.

Download (ext.)

https://arxiv.org/abs/1402.4489 [Preprint]

DOI

10.1515/crelle-2014-0122

Journal Article | Submitted | English

Author

Browning, Timothy D^ISTA ; Prendiville, Sean

Abstract

We show that a non-singular integral form of degree d is soluble non-trivially over the integers if and only if it is soluble non-trivially over the reals and the p-adic numbers, provided that the form has at least (d-\sqrt{d}/2)2^d variables. This improves on a longstanding result of Birch.

Publishing Year

2015

Date Published

2015-02-20

Journal Title

Journal fur die Reine und Angewandte Mathematik

Publisher

Walter de Gruyter

Acknowledgement

While working on this paper the authors were supported by the Leverhulme Trust and ERC grant 306457.

Volume

2017

Issue

731

Page

203 - 234

ISSN

0075-4102

IST-REx-ID

271

Cite this

Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many variables. Journal fur die Reine und Angewandte Mathematik. 2017(731):203-234. doi:10.1515/crelle-2014-0122

Browning, T. D., & Prendiville, S. (n.d.). Improvements in Birch’s theorem on forms in many variables. Journal Fur Die Reine Und Angewandte Mathematik. Walter de Gruyter. https://doi.org/10.1515/crelle-2014-0122

Browning, Timothy D, and Sean Prendiville. “Improvements in Birch’s Theorem on Forms in Many Variables.” Journal Fur Die Reine Und Angewandte Mathematik. Walter de Gruyter, n.d. https://doi.org/10.1515/crelle-2014-0122.

T. D. Browning and S. Prendiville, “Improvements in Birch’s theorem on forms in many variables,” Journal fur die Reine und Angewandte Mathematik, vol. 2017, no. 731. Walter de Gruyter, pp. 203–234.

Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many variables. Journal fur die Reine und Angewandte Mathematik. 2017(731), 203–234.

Browning, Timothy D., and Sean Prendiville. “Improvements in Birch’s Theorem on Forms in Many Variables.” Journal Fur Die Reine Und Angewandte Mathematik, vol. 2017, no. 731, Walter de Gruyter, pp. 203–34, doi:10.1515/crelle-2014-0122.

All files available under the following license(s):

Copyright Statement: