Improvements in Birch's theorem on forms in many variables
  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. 
T.D. Browning and Sean Prendiville
math.NT