Properties of Ascending Chains for Partial Difference Polynomial Systems

A characteristic set theory for partial difference polynomial systems is proposed. We introduce the concept of coherent and regular ascending chains and prove that a partial difference ascending chain is the characteristic set of its saturation ideal if and only if it is coherent and regular. This gives a method to decide whether a polynomial belongs to the saturation ideal of an ascending chain. We introduce the concept of strongly irreducible ascending chains and prove that a partial difference ascending chain is the characteristic set of a reflexive prime ideal if and only if it is strongly irreducible. This gives a simple and precise representation for reflexive prime ideals.

[1]  Xiao-Shan Gao,et al.  Ritt-Wu's Decomposition Algorithm and Geometry Theorem Proving , 1990, CADE.

[2]  Wenjun Wu,et al.  Mechanical Theorem Proving in Geometries , 1994, Texts and Monographs in Symbolic Computation.

[3]  Dongming Wang,et al.  Elimination Methods , 2001, Texts and Monographs in Symbolic Computation.

[4]  M. M. Maza,et al.  Well known theorems on triangular systems and the D5 principle , 2006 .

[5]  François Boulier,et al.  Representation for the radical of a finitely generated differential ideal , 1995, ISSAC '95.

[6]  A. V. Mikhalev,et al.  Differential and Difference Dimension Polynomials , 1998 .

[7]  H. W. Raudenbush,et al.  Ideal theory and algebraic difference equations , 1934 .

[8]  Marc Moreno Maza,et al.  On the Theories of Triangular Sets , 1999, J. Symb. Comput..

[9]  Evelyne Hubert,et al.  Factorization-free Decomposition Algorithms in Differential Algebra , 2000, J. Symb. Comput..

[10]  S. Chou.  Mechanical Geometry Theorem Proving , 1987 .

[11]  Tun Wen-Dun.  ON THE DECISION PROBLEM AND THE MECHANIZATION OF THEOREM-PROVING IN ELEMENTARY GEOMETRY , 1978 .

[12]  Yong Luo,et al.  A characteristic set method for ordinary difference polynomial systems , 2009, J. Symb. Comput..

[13]  A. Rosenfeld.  Specializations in differential algebra , 1959 .

[14]  Irving Bentsen.  The existence of solutions of abstract partial difference polynomials. , 1971 .

[15]  Xiao-Shan Gao,et al.  Complete numerical isolation of real zeros in zero-dimensional triangular systems , 2007, ISSAC '07.

[16]  Hamid Maarouf,et al.  Unmixed-dimensional Decomposition of a Finitely Generated Perfect Differential Ideal , 2001, J. Symb. Comput..

[17]  Marc Moreno Maza,et al.  Lifting techniques for triangular decompositions , 2005, ISSAC.

[18]  Xiao-Shan Gao,et al.  A Zero Structure Theorem for Differential Parametric Systems , 1993, J. Symb. Comput..

[19]  Joris van der Hoeven,et al.  Characteristic set method for differential-difference polynomial systems , 2009, J. Symb. Comput..

[20]  P. Aubry,et al.  Ensembles triangulaires de polynomes et resolution de systemes algebriques. Implantation en axiom , 1999 .