An improved and simplified binary Ax theorem

Chevalley´s theorem establishing conditions for the existence of solutions of a system of polynomial equations over a finite field F_q of characteristic p, was refined by Warning an Ax by the introduction of divisibility properties on the number of solutions. These classical result depends strongly on the total degree of the system. For the binary case we show improvements of Chevallev-Warning and Ax theorems in two directions. First we obtain more accurate divisibility of the number of zeros of the system in a result completely independent of the system degree. Secondly, the proof is of strict elementary combinatorial nature, contrary to Ax´s proof which depends on advanced methods of Gaussian sums and p-adic evaluations. The method is applied to Reed-Muller codes