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

TL;DR

This paper explores the connection between the number of solutions of polynomial congruences and the resultant of two polynomials, providing new proofs for known Lucas sequence congruences.

Contribution

It establishes a relationship between the solutions of polynomial systems modulo a prime and the polynomial resultant, offering new proofs for Lucas sequence congruences.

Findings

The number of solutions of the system influences the divisibility of the resultant.

The result applies to polynomials with integer coefficients modulo an odd prime.

New proofs of known Lucas sequence congruences are derived.

Abstract

Let $q$ be an odd prime and $f(x)$, $g(x)$ be polynomials with integer coefficients. If the system of congruences $f(x) \equiv g(x) \equiv 0 \pmod{q}$ has $\ell$ solutions, then $R\left(f(x),g(x)\right)\equiv 0 \pmod{q^\ell}$, 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.