Characteristic decomposition of polynomial sets

Abstract

An ordered pair $(\pset{G},\,~\pset{C})$ is called the characteristic pair of a polynomial ideal if $\pset{G}$ is the reduced lexicographic grobner basis of the ideal and $\pset{C}$ is the minimal triangular set contained in $\pset{G}$. A characteristic pair $(\pset{G},\,~\pset{C})$ is said to be regular or normal if $\pset{C}$ is regular or normal; it is said to be strong if the ideal generated by $\pset{G}$ and the saturated ideal of $\pset{C}$ are the same. A strong regular or normal characteristic decomposition of a polynomial set means a decomposition of the polynomial set into finitely many (strong) regular or normal characteristic pairs with an associated zero or ideal decomposition. This paper presents two algorithms for characteristic decomposition of any given polynomial set, one producing strong normal characteristic pairs with ideal splitting according to pseudo-divisibility relations between polynomials in reduced lexicographic grobner bases and the other producing strong regular characteristic pairs by constructing nontrivial ideals that can be divided out of given ideals by using ideal quotient. Novel theorems are established for regularizing characteristic pairs and strengthening regular and normal characteristic pairs by using ideal saturation. Strong regular or normal characteristic pairs produced by the algorithms possess nice properties and furnish two representations, one in terms of lexicographic grobner bases and the other in terms of triangular sets, for the zeros of the input set of multivariate polynomials. The paper includes a brief review on triangular decompositions of various kinds and discussions about inherent connections among triangular sets, Ritt characteristic sets, and reduced lexicographic grobner bases. Examples and experimental results are also provided to illustrate the method of characteristic decomposition and its applicability and performance.