Improvements in Birch's theorem on forms in many variables

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.

36 pages

References in corpus (2)

• On the destruction of minimal foliations
• Birch's theorem in function fields

Cited by in corpus (1)

• Sarnak's saturation problem for complete intersections