Maximum number of common zeros of homogeneous polynomials over finite fields

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the the maximum number of common zeros that r linearly independent homogeneous polynomials of degree d in m + 1 variables with coefficients in a finite field with q elements can have in the corresponding m-dimensional projective space over that finite field.

Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if r is at most m + 1 and can be invalid otherwise. Moreover a new conjecture was proposed for many values of r beyond m + 1. In this paper, we prove that this new conjecture holds true for several values of r. In particular, this settles the new conjecture completely when d = 3.

Our result also includes the positive result of Datta and Ghorpade as a special case. Further, we also determine the maximum number of zeros in certain cases not covered by the earlier conjectures and results, namely, the case of d = q − 1 and of d = q.