Improvements in Birch's theorem on forms in many variables

Tim D Browning; Sean M Prendiville

doi:10.1515/crelle-2014-0122

Improvements in Birch's theorem on forms in many variables

Tim D Browning, Sean M Prendiville

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Abstract

We show that a non-singular integral form of degree d is soluble over the integers if and only if it is soluble over ℝ and over ℚ_p for all primes p, provided that the form has at least (d - d^1/2/2)2^d variables. This improves on a longstanding result of Birch.

Original language	English
Pages (from-to)	203-234
Number of pages	32
Journal	Journal für die reine und angewandte Mathematik
Volume	2017
Issue number	731
Early online date	20 Feb 2015
DOIs	https://doi.org/10.1515/crelle-2014-0122
Publication status	Published - 1 Oct 2017

Keywords

math.NT
11P55 (11G35, 14G05)

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AU - Prendiville, Sean M

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AB - We show that a non-singular integral form of degree d is soluble over the integers if and only if it is soluble over ℝ and over ℚp for all primes p, provided that the form has at least (d - d1/2/2)2d variables. This improves on a longstanding result of Birch.

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DO - 10.1515/crelle-2014-0122

M3 - Article (Academic Journal)

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JO - Journal für die reine und angewandte Mathematik

JF - Journal für die reine und angewandte Mathematik

IS - 731

ER -

Improvements in Birch's theorem on forms in many variables

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