Improvements in Birch's theorem on forms in many variables - Browning, Timothy D; Prendiville, Sean (Journal fur die Reine und Angewandte Mathematik, 2015)
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.
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