Improvements in Birch's theorem on forms in many variables - Browning, Timothy D; Prendiville, Sean (Journal fur die Reine und Angewandte Mathematik, 2017)
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 - 1/2 √d)2d variables. This improves on a longstanding result of Birch.
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