Improvements in Birch’s theorem on forms in many variables

Abstract

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℝ</m:mi> </m:math> {\mathbb{R}} and over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℚ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Q}_{p}} for all primes p , provided that the form has at least <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mo>-</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> <m:mo>⁢</m:mo> <m:msqrt> <m:mi>d</m:mi> </m:msqrt> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mn>2</m:mn> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> {(d-\frac{1}{2}\sqrt{d})2^{d}} variables. This improves on a longstanding result of Birch.