Ideals Modulo a Prime

Abstract

The main focus of this paper is on the problem of relating an ideal [Formula: see text] in the polynomial ring [Formula: see text] to a corresponding ideal in [Formula: see text] where [Formula: see text] is a prime number; in other words, the reduction modulo[Formula: see text] of [Formula: see text]. We first define a new notion of [Formula: see text]-good prime for [Formula: see text] which does depends on the term ordering [Formula: see text], but not on the given generators of [Formula: see text]. We relate our notion of [Formula: see text]-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 [Formula: see text] from the term ordering, thus letting us show that all but finitely many primes are good for [Formula: see text]. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.