A new algorithmic scheme for computing characteristic sets

Abstract

Ritt-Wu's algorithm of characteristic sets is the most representative for triangularizing sets of multivariate polynomials. Pseudo-division is the main operation used in this algorithm. In this paper we present a new algorithmic scheme for computing generalized characteristic sets by introducing other admissible reductions than pseudo-division. A concrete subalgorithm is designed to triangularize polynomial sets using selected admissible reductions and several effective elimination strategies and to replace the algorithm of basic sets (used in Ritt-Wu's algorithm).The proposed algorithm has been implemented and experimental results show that it performs better than Ritt-Wu's algorithm in terms of computing time and simplicity of output for a number of non-trivial test examples.

Complex Dynamics of Knowledgeable Monopoly Models with Gradient Mechanisms

“…By using the method of triangular decomposition 1 , we transform the solutions of the first two equations of system (12) into zeros of the triangular set…”

“…However, the triangular decomposition method is available for polynomial systems, while the Gaussian elimination method is just for linear systems. Refer to[30,16,12,29] for more details.…”

Chordal graphs in triangular decomposition in top-down style

“…Following the above terminologies, the conclusions of Theorem 12 can be reformulated as: G(red n (P)) ⊂ G(P), and the equality holds if supp(T n ) = supp(P (n) ). Indeed, the reduction process above is commonly used in algorithms for triangular decomposition in top-down style, and the mapping f i in ( 2) is abstraction of specific reductions used in different kinds of algorithms for triangular decomposition [25]. For example, one specific kind of such reduction is performed by using pseudo divisions, and in this case R in (2) consists of pseudo remainders which do not contain x i .…”

On the Chordality of Polynomial Sets in Triangular Decomposition in Top-Down Style

“…Remark 3.1 Note that red 1 (P) forms a triangular set after reordering if red 1 (P) does not contain any non-zero constant. Indeed, the reduction process to compute this triangular set is commonly used in algorithms for triangular decomposition in top-down style, and the mapping f i in (2) is abstraction of specific reductions used in different kinds of algorithms for triangular decomposition [20]. For example, one specific kind of such reduction is performed by using pseudo-divisions, and in this case R in (2) consists of pseudo-remainders which do not contain x i .…”