Decomposing Polynomial Sets Simultaneously into Gröbner Bases and Normal Triangular Sets

In this paper we focus on the algorithms and their implementations for decomposing an arbitrary polynomial set simultaneously into pairs of lexicographic Gröbner bases and normal triangular sets with inherent connections in between and associated zero relationship with the polynomial set. In particular, a method by temporarily changing the variable orderings to handle the failure of the variable ordering assumption is proposed to ensure splitting needed for characteristic decomposition. Experimental results of our implementations for (strong) characteristic decomposition with comparisons with available implementations of triangular decomposition are also reported.