Ideals modulo a prime

Abstract

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb{Q}[x_1,\dots ,x_n]$ to a corresponding ideal in ${\mathbb{F}}_p[x_1,\dots ,x_n]$ where $p$ is a prime number; in other words, the \textit{reduction modulo $p$} of $I$. We first define a new notion of $\sigma$-good prime for $I$ which does depends on the term ordering $\sigma$, but not on the given generators of $I$. We relate our notion of $\sigma$-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo~$p$ from the term ordering, thus letting us show that all but finitely many primes are good for $I$. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

Improvements, and extended bibliography. To be published on "Journal of Algebra and Its Applications (JAA)"