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–Lindström Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed–Muller codes are also included.