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