Strategies for Polynomial Systems

Abstract

Before we devote numerous pages to the topic of polynomial systems of equations over finite fields, we should pause and ask why one would want to do this. This will give us an opportunity to highlight the several applications of this interesting area.

Before that, however, an important distinction must be made. In this book, when one has a system of equations over the finite field $\mathbb{G}\mathbb{F}$(pⁿ), we assume that one is interested only in those solutions which are also elements of $\mathbb{G}\mathbb{F}$(pⁿ). If one is also interested in solutions in some extension field, (e.g. $\mathbb{G}\mathbb{F}$(p^m) with n|m), then since $\mathbb{G}\mathbb{F}$(pⁿ) is a subfield of $\mathbb{G}\mathbb{F}$(p^m), it is safe to consider the system of equations as if it were over $\mathbb{G}\mathbb{F}$(pⁿ), or at worse the splitting field.