On the relationship between the number of solutions of congruence systems and the resultant of two polynomials

Abstract

Let $q$ be an odd prime and $f(x)$, $g(x)$ be polynomials with integer coefficients. If the system of congruences $f(x)\equiv g(x)\equiv 0(\mathrm{mod}q)$ has $\ell$ solutions, then $R\left(f(x),g(x)\right)\equiv 0(\mathrm{mod}{q}^{\ell })$, where $R\left(f(x),g(x)\right)$ is the resultant of the polynomials. Using this result we give new proofs of some known congruences involving the Lucas sequences.