A generalized criterion for signature related Gröbner basis algorithms

  • Authors:

  • Yao Sun;Dingkang Wang

  • Affiliations:

  • Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China;Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

  • Venue:
  • Proceedings of the 36th international symposium on Symbolic and algebraic computation
  • Year:
  • 2011

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Abstract

A generalized criterion for signature related algorithms to compute Gröbner basis is proposed in this paper. Signature related algorithms are a popular kind of algorithms for computing Gröbner basis, including the famous F5 algorithm, the F5C algorithm, the extended F5 algorithm and the GVW algorithm. The main purpose of current paper is to study in theory what kind of criteria is correct in signature related algorithms and provide a generalized method to develop new criteria. For this purpose, a generalized criterion is proposed. The generalized criterion only relies on a general partial order defined on a set of polynomials. When specializing the partial order to appropriate specific orders, the generalized criterion can specialize to almost all existing criteria of signature related algorithms. For admissible partial orders, a proof is presented for the correctness of the algorithm that is based on this generalized criterion. And the partial orders implied by the criteria of F5 and GVW are also shown to be admissible in this paper. More importantly, the generalized criterion provides an effective method to check whether a new criterion is correct as well as to develop new criteria for signature related algorithms.