About two decades ago, Tsfasman and Boguslavsky conjectured a formula for thethe maximum number of common zeros that r linearly independent homogeneous polynomialsof degree d in m + 1 variables with coefficients in a finite field with q elements can have inthe corresponding m-dimensional projective space over that finite field. Recently, it has beenshown by Datta and Ghorpade that this conjecture is valid if r is at most m + 1 and canbe invalid otherwise. Moreover a new conjecture was proposed for many values of r beyondm + 1. In this paper, we prove that this new conjecture holds true for several values of r. Inparticular, this settles the new conjecture completely when d = 3. Our result also includesthe positive result of Datta and Ghorpade as a special case. Further, we also determine themaximum 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.
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