Ideals modulo a prime

TL;DR

This paper introduces a new framework for understanding how polynomial ideals over rationals relate to their reductions modulo primes, including a novel notion of good primes and an invariant called the universal denominator.

Contribution

It defines a new concept of -good primes , independent of generators, and introduces the universal denominator invariant to identify primes suitable for modular methods.

Findings

Most primes are good for a given ideal.

The universal denominator helps detect bad primes.

The approach simplifies modular computations in polynomial ideal theory.

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.