Title: Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields,
Abstract: About two decades ago, Tsfasman and Boguslavsky conjectured a formula for 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. 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 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 $d=q$. All these results are directly applicable to the determination of the
maximum number of points on sections of Veronese varieties by linear
subvarieties of a fixed dimension, and also the determination of generalized
Hamming weights of projective Reed-Muller codes.