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.