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arXiv · 2102.00602

Elementary analysis of isolated zeroes of a polynomial system

Mitali Bafna·Madhu Sudan·Santhoshini Velusamy·David Xiang
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Abstract

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 $\mathbb{F}$.

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Mitali Bafna, Madhu Sudan, Santhoshini Velusamy, David Xiang. 2021-02-01. Elementary analysis of isolated zeroes of a polynomial system. https://arxiv.org/abs/2102.00602

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