Decomposing the Radicals of Polynomial Ideals by Rational Univariate Representations

In this paper,the notion of rational univariate representations with variables is introduced.Consequently,the ideals,created by given rational univariate representations with variables,are defined.One merit of these created ideals is that some of their algebraic properties can be easily decided.With the aid of the theory of valuations,some related results are established.Based on these results,a new approach is presented for decomposing the radical of a polynomial ideal into an intersection of prime ideals.

The Invertibility of Rational Univariate Representations

In this paper,the so-called invertibility is introduced for rational univariate representations,and a characterization of the invertibility is given.It is shown that the rational univariate representations,obtained by both Rouillier’s approach and Wu’s method,are invertible.Moreover,the ideal created by a given rational univariate representation is defined.Some results on invertible rational univariate representations and created ideals are established.Based on these results,a new approach is presented for decomposing the radical of a zero-dimensional polynomial ideal into an intersection of maximal ideals.

Computation of the Rational Representation for Solutions of High-dimensional Systems

This paper deals with the representation of the solutions of a polynomial system, and concentrates on the high-dimensional case. Based on the rational univari- ate representation of zero-dimensional polynomial systems, we give a new description called rational representation for the solutions of a high-dimensional polynomial sys- tem and propose an algorithm for computing it. By this way all the solutions of any high-dimensional polynomial system can be represented by a set of so-called rational- representation sets.

A Simplified Rational Representation for Positive-Dimensional Polynomial Systems and SHEPWM Equations Solving

The paper is concerned with the improvement of the rational representation theory for solving positive-dimensional polynomial systems. The authors simplify the expression of rational representation set proposed by Tan and Zhang(2010), obtain the simplified rational representation with less rational representation sets, and hence reduce the complexity for representing the variety of a positive-dimensional ideal. As an application, the authors compute a "nearly" parametric solution for the SHEPWM problem with a fixed number of switching angles.

An Improvement of the Rational Representation for High-Dimensional Systems

Based on the rational univariate representation of zero-dimensional polynomial systems,Tan and Zhang proposed the rational representation theory for solving a high-dimensional polynomial system,which uses so-called rational representation sets to describe all the zeros of a high-dimensional polynomial system.This paper is devoted to giving an improvement for the rational representation.The idea of this improvement comes from a minimal Dickson basis used for computing a comprehensive Grobner system of a parametric polynomial system to reduce the number of branches.The authors replace the normal Grobner basis G satisfying certain conditions in the original algorithm(Tan-Zhang’s algorithm)with a minimal Dickson basis G_(m) of a Grobner basis for the ideal,where G_(m) is smaller in size than G.Based on this,the authors give an improved algorithm.Moreover,the proposed algorithm has been implemented on the computer algebra system Maple.Experimental data and its performance comparison with the original algorithm show that it generates fewer branches and the improvement is rewarding.

Algorithms for computing the global infimum and minimum of a polynomial function

By catching the so-called strictly critical points,this paper presents an effective algorithm for computing the global infimum of a polynomial function.For a multivariate real polynomial f ,the algorithm in this paper is able to decide whether or not the global infimum of f is finite.In the case of f having a finite infimum,the global infimum of f can be accurately coded in the Interval Representation.Another usage of our algorithm to decide whether or not the infimum of f is attained when the global infimum of f is finite.In the design of our algorithm,Wu’s well-known method plays an important role.