Elementary analysis of isolated zeroes of a polynomial system

Wooley (J. Number Theory, 1996) gave an elementary proof of a Bezout like theorem allowing one to count the number of isolated integer roots of a system of polynomial equations modulo some prime power. In this article, we adapt the proof to a slightly different setting. Specifically, we consider polynomials with coefficients from a polynomial ring F[t] for an arbitrary field F and give an upper bound on the number of isolated roots modulo ts for an arbitrary positive integer s. In particular, using s = 1, we can bound the number of isolated roots of a system of polynomials over an arbitrary field F.