A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible $\mathbb{F}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field $\mathbb{F}q$ with $q$ elements, when $d<q$. It is shown that this formula holds in the affirmative for several values of $r$. In the general case, we give explicit lower and upper bounds for $e_r(d,m)$ and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindstr"om Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.