- Let f1, . . . , fr be polynomials in n variables, over the field Fq, and suppose that their degrees are d1, . . . , dr. It was shown by Warning in 1935 that if N is the number of common zeros of the polynomials fi, then N > qn?d. 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 Fnq , it is shown for example that N > 2qn?d if q ≥ 4, and that N ≥ qn+1?d/(n + 2 ? d) if the fi are all homogeneous. © 2011 RAS(DoM) and LMS.
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