Ideals Modulo a Prime

Abstract

The main focus of this paper is on the problem of relating an ideal I in the polynomial ring Q[x 1 , . . . , xn] to a corresponding ideal in Fp[x 1 , . . . , xn] where p is a prime number; in other words, the reduction modulo p of I. We first define a new notion of σ-good prime for I which does depends on the term ordering σ, but not on the given generators of I. We relate our notion of σ-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.

Computing and using minimal polynomials

“…On the theoretical side, we investigate more deeply the consequences of our new modular approach and apply it to general ideals in the preprint by Abbott et al (2018b).…”

“…In the literature, several authors have looked at various notions of bad reduction in similar contexts, see for instance the articles by Noro and Yokoyama (2018), Pauer (2007), Winkler (1988) and Arnold (2003). However, our approach is systematically tied to reduced Gröbner bases as computationally robust representations of ideals; we study more deeply this approach in the preprint (Abbott et al, 2018b). The combination of the theoretical results explained in this section with various implementation details lead to the good practical performance as shown in Table 2.…”

Modular Techniques for Intermediate Primary Decomposition

“…Since the timing of the full primary decomposition of 𝑐𝑦𝑐𝑙𝑖𝑐 (6) is 148 seconds, both methods have effectiveness by its speciality. Here, 𝑐𝑦𝑐𝑙𝑖𝑐 (4) is a mixed 1-dimensional ideal and the strong intermediate prime decomposition is We note that 𝑐𝑦𝑐𝑙𝑖𝑐 (4) [2] 𝐺 has the same strong intermediate prime decomposition. However, the full primary decomposition of 𝑐𝑦𝑐𝑙𝑖𝑐 (4) [2] 𝐺 is harder to compute (see Table 3) as there are intermediate coefficient growth and some primary components are very complicated.…”

“…This uniqueness is useful for reconstructing modular results to the rational field in Algorithm 2 in Section 3. 2 ⟩} is a primary decomposition of 𝐼 and Ass(𝐼 ) = {⟨𝑥, 𝑦⟩, ⟨𝑦, 𝑧⟩, ⟨𝑧, 𝑥⟩, ⟨𝑥, 𝑦, 𝑧⟩}. Here, ⟨𝑥, 𝑦⟩, ⟨𝑦, 𝑧⟩ and ⟨𝑧, 𝑥⟩ are isolated prime divisors of 𝐼 and ⟨𝑥, 𝑦, 𝑧⟩ is the embedded prime divisor of 𝐼 .…”

“…The modular method has been applied to computations of Gröbner basis, ideal quotient, saturation, and so on [1,2,5,8,21,23]. In this paper, we apply it to computation of intermediate primary decomposition.…”