The main focus of this paper is on the problem of relating an ideal I in the polynomial ring Q[x_1,..., x_n] to a corresponding ideal in F_p[x_1, ..., x_n] where p is a prime number; in other words, the reduction modulo p of I. We define a new notion of sigma-good prime for I which depends on the term ordering sigma, and show that all but finitely many primes are good for all term orderings. We relate our notion of sigma-good primes to some other similar notions already in the literature. One characteristic of our approach is that enables us to detect some bad primes, a distinct advantage when using modular methods.

Ideals modulo p

John Abbott;Anna Maria Bigatti;Lorenzo Robbiano
2018

Abstract

The main focus of this paper is on the problem of relating an ideal I in the polynomial ring Q[x_1,..., x_n] to a corresponding ideal in F_p[x_1, ..., x_n] where p is a prime number; in other words, the reduction modulo p of I. We define a new notion of sigma-good prime for I which depends on the term ordering sigma, and show that all but finitely many primes are good for all term orderings. We relate our notion of sigma-good primes to some other similar notions already in the literature. One characteristic of our approach is that enables us to detect some bad primes, a distinct advantage when using modular methods.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/898677
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