Improvements in Birch's theorem on forms in many variables

2014/02/18 by T. D. Browning, Browning, T. D., Sean Prendiville +1
Mathematics · #11P55 (11G35 #14G05) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1402.4489

openalex publication_date 2014/02/18 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28

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-√(d)/2)2^d variables. This improves on a longstanding result of Birch.

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