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arXiv · 1510.07330

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

Dmitry I. Khomovsky
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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.

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Dmitry I. Khomovsky. 2016-10-10. On the relationship between the number of solutions of congruence systems and the resultant of two polynomials. https://arxiv.org/abs/1510.07330

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