Regular chain

In computer algebra, a regular chain is particular kind of triangular set in a multivariate polynomial ring over a field, which is a fine generalization of the notion of characteristic set.

Introduction

Given a linear system, one can convert it to a triangular system via Gaussian elimination. For the non-linear case, given a polynomial system F over a field, one can convert it to a finite set of triangular sets, in the sense that the algebraic variety V(F) is described by these triangular sets. Characteristic sets are triangular sets of such kind. A characteristic set may merely describe an empty set. We call it the degenerated case. To fix this, the notion of regular chain was introduced, independently by Kalkbrener (1993), Yang and Zhang (1994). A regular chain always describes a nonempty unmixed variety.

Formal definitions

   -> initial, .....
   -> characteristic set, triangular set, reg chain, ....
       (see the paper "On the theories....")
   -> triangular decomp (Lazard and Kalkbrener senses)
   -> regular system, constructible set

o Properties and Applications

   -> Properties (the T=Sat(T)) paper
   -> applications: see the RC lib, diff algebra

o Algorithms and implementation (with links)

   -> Wu's method, Wsolve, Discoverer, AXIOM, Singular, ...
   -> DynamicEvaluation
   -> Epsilon
   -> Kalkbrener
   -> LexTriangular (Singular)
   -> Triade

o History

   See "On the theories of ..."

o Related links

    -> RC lib