Solvability of systems of polynomial equations with multivariate polynomials as coefficients

Abstract

In [3] Castro, Moreno and Rubio generalize the results of Moreno-Moreno's theorem that gives a bound for the power of a prime <em>p</em> to divide the number of common zeros of the multivariate polynomials <em>F₁</em>,...,<em>F_t</em> over a finite field. This generalization regarded the coefficients of the polynomials to be uni-variate polynomials over a finite field instead of plain elements of the finite field. The result led to improve a theorem of Carlitz, for the estimation of the number of variables needed so that a system of polynomial equations with coefficients in <em>F_q</em>[<em>X</em>] can have non-trivial zeros. We generalize the results of Castro, Moreno and Rubio to polynomials whose coefficients are multivariate polynomials over finite fields.