多项式系统孤立零点的初等分析
Elementary analysis of isolated zeroes of a polynomial system
摘要：
Wooley（{ \em J.数论}，1996）给出了Bezout定理的初等证明，该定理允许计算多项式方程模某个素数幂的系统的孤立整数根的个数。在本文中，我们将证明调整到稍微不同的设置。具体地说，我们考虑多项式的系数从一个多项式环$\mathbb{F}[t]$为任意域$\mathbb{F}$和给定的上界上的孤立根模$t^s$的数目为任意正整数$s$。特别地，使用$s=1$，我们可以限制任意域$\mathbb{F}$上的多项式系统的孤立根的数目。
英文摘要：
Wooley ({\em 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 $\mathbb{F}[t]$ for an arbitrary field $\mathbb{F}$ and give an upper bound on the number of isolated roots modulo $t^s$ 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 $\mathbb{F}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
