Computing Triangular Systems and Regular Systems

Abstract

International audienceA previous algorithm of computing simple systems is modified and extended to compute triangular systems and regular systems from any given polynomial system. The resulting algorithms, based on the computation of subresultant regular subchains, have a simple structure and are efficient in practice. Preliminary experiments indicate that they perform at least as well as some of the known algorithms. Several properties about regular systems are also proved

On the Regularity Property of Differential Polynomials Modulo Regular Differential Chains

“…Theorem 1 enhances also [18,Proposition 5.3]. Indeed, this Proposition states that res(f, C) = 0 if, and only if, the polynomial f does not annihilate on any "regular zero" of C, where "regular zeros" are defined as generic zeros of the associated prime ideals of B which have dimension |N | and do not meet the multiplicative family M generated by the nonzero elements of K[N ] (see [18,Definition 5.1]).…”

“…x k . The next lemma is borrowed from [18,Lemma 5.2]. Together with the two following ones, it prepares the proof of Theorem 1.…”

“…Indeed, this Proposition states that res(f, C) = 0 if, and only if, the polynomial f does not annihilate on any "regular zero" of C, where "regular zeros" are defined as generic zeros of the associated prime ideals of B which have dimension |N | and do not meet the multiplicative family M generated by the nonzero elements of K[N ] (see [18,Definition 5.1]). However, [18] misses [5,Theorem 1.6] which states that this property is held by all the associated prime ideals of B. See the comments on [6,Algorithm Invert].…”

“…However, the proof of that Theorem just refers to [8] and [18] and thus misses the use of [5,Theorem 1.6].…”

Algebraic Approaches to Stability Analysis of Biological Systems

“…In the case where r = n, we have x pi = x i . Effective algorithms have been developed by Wu [49], Lazard [23], Kalkbrener [19], and Wang [40,41,42] to decompose any polynomial set P or system [P, Q] (i.e., a pair of polynomial sets, where…”