The RegularChains library in MAPLE

Abstract

The standard approach for computing with an algebraic number is through the data of its irreducible minimal polynomial over some base field k. However, in typical tasks such as polynomial system solving, involving many algebraic numbers of high degree, following this approach will require using probably costly factorization algorithms. Della Dora, Dicrescenzo and Duval introduced "dynamic evaluation" techniques (also termed "D5 principle") [3] as a means to compute with algebraic numbers, while avoiding factorization. Roughly speaking, this approach leads one to compute over direct products of field extensions of k, instead of only field extensions.

Doing algebraic geometry with the RegularChains library

“…Today, regular chains are at the core of algorithms computing triangular decomposition of polynomial systems and which are available in several software packages [7,9,10]. Moreover, those algorithms provide back-engines for computer algebra system front-end solvers, such as Maple's solve command.…”

Chemical Reaction Systems, Computer Algebra and Systems Biology

“…Computation using MABSys. The same computation can also be performed by the dedicated MABSys package [24,23], which relies on the MAPLE RegularChains [21] and ELPS [36] packages. The MABSys package gathers as input a chemical reaction system.…”

Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library

“…We refer to [1,6,5] for the notion of a regular chain, its related concepts, such as initial, saturated ideals, quasi-components and the related operations.…”

“…heuristically-with-split: It tests the conditions (C 1 ) and (C h ). Checking (C 1 ) is achieved by means of the operation Regularize [5,6]: for a polynomial p and a regular chain T , Regularize(p, T ) returns regular chains T 1 , . .…”

“…Then, based on these tools, we obtain several algorithms for removing redundant components in a triangular decomposition. We report on the implementation of these different solutions within the RegularChains library [5].…”