Some notes upon "When does < T > equal sat(T)?"

Abstract

Given a regular set T in K[x], Lemaire et al. in ISSAC'08 give a nice algebraic property: the regular set T generates its saturated ideal if and only if it is primitive. We firstly aim at giving a more direct proof of the above result, generalizing the concept of primitivity of polynomials and regular sets and presenting a new result which is equivalent to the above property. On the other hand, based upon correcting an error of the definition of U-set in AISC'06, we further develop some geometric properties of triangular sets. To a certain extent, the relation between the primitivity of T and its U-set is also revealed in this paper. © 2010 Springer-Verlag Berlin Heidelberg.