A note on the Chevalley-Warning theorems

Let be polynomials in variables, over the field , and suppose that their degrees are . It was shown by Warning in 1935 that if is the number of common zeros of the polynomials , then . It is the main aim of the present paper to improve on this bound. When the set of common zeros does not form an affine linear subspace in , it is shown for example that if , and that if the are all homogeneous.