arXiv subjects

Dingkang Wang

Publications and source records attributed to Dingkang Wang.

26 records · Page 2Linked to original sources

arXiv2014

An Improvement over the GVW Algorithm for Inhomogeneous Polynomial Systems

The GVW algorithm is a signature-based algorithm for computing Gröbner bases. If the input system is not homogeneous, some J-pairs with higher signatures but lower degrees are rejected by GVW's Syzygy Criterion, instead, GVW have to compute some J-pairs with lower signatures but higher degrees. Consequently, degrees of polynomials appearing during the computations may unnecessarily grow up higher and the computation become more expensive. In this paper, a variant of the GVW algorithm, called M-GVW, is proposed and mutant pairs are introduced to overcome inconveniences brought by inhomogeneous input polynomials. Some techniques from linear algebra are used to improve the efficiency. Both GVW and M-GVW have been implemented in C++ and tested by many examples from boolean polynomial rings. The timings show M-GVW usually performs much better than the original GVW algorithm when mutant pairs are found. Besides, M-GVW is also compared with intrinsic Gröbner bases functions on Maple, Singular and Magma. Due to the efficient routines from the M4RI library, the experimental results show that M-GVW is very efficient.

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arXiv2011

Solving Detachability Problem for the Polynomial Ring by Signature-based Groebner Basis Algorithms

Signature-based algorithms are a popular kind of algorithms for computing Groebner basis, including the famous F5 algorithm, F5C, extended F5, G2V and the GVW algorithm. In this paper, an efficient method is proposed to solve the detachability problem. The new method only uses the outputs of signature-based algorithms, and no extra Groebner basis computations are needed. When a Groebner basis is obtained by signature-based algorithms, the detachability problem can be settled in polynomial time.

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arXiv2011

A Generalized Criterion for Signature-based Algorithms to Compute Gröbner Bases

A generalized criterion for signature-based algorithms to compute Gröbner bases is proposed in this paper. This criterion is named by "generalized criterion", because it can be specialized to almost all existing criteria for signature-based algorithms which include the famous F5 algorithm, F5C, extended F5, G$^2$V and the GVW algorithm. The main purpose of current paper is to study in theory which kind of criteria is correct in signature-based algorithms and provide a generalized method to develop new criteria. For this purpose, by studying some key facts and observations of signature-based algorithms, a generalized criterion is proposed. The generalized criterion only relies on a 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-based algorithms. For {\em 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. 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-based algorithms.

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arXiv2011

A Generalized Criterion for Signature Related Gröbner Basis Algorithms

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 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 {\em admissible} partial orders, a complete proof of the correctness of the algorithm based on this generalized criterion is also presented. This proof has no extra requirements on the computing order of critical pairs, and is also valid for non-homogeneous polynomial systems. More importantly, the partial orders implied by existing criteria are admissible. Besides, one can also check whether a new criterion is correct in signature related algorithms or even develop new criteria by using other admissible partial orders in the generalized criterion.

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arXiv2010

The F5 Algorithm in Buchberger's Style

The famous F5 algorithm for computing \gr basis was presented by Faugère in 2002. The original version of F5 is given in programming codes, so it is a bit difficult to understand. In this paper, the F5 algorithm is simplified as F5B in a Buchberger's style such that it is easy to understand and implement. In order to describe F5B, we introduce F5-reduction, which keeps the signature of labeled polynomials unchanged after reduction. The equivalence between F5 and F5B is also shown. At last, some versions of the F5 algorithm are illustrated.

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arXiv2010

An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field

A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra Groebner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a polynomial over the ground field by a generic linear map. Then this polynomial is factorized over the ground field. From these factors, the factorization of the polynomial over the extension field is obtained. The new algorithm has been implemented and computer experiments indicate that the new algorithm is very efficient, particularly in complicated examples.

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arXiv2010

On Computing Groebner Basis in the Rings of Differential Operators

Insa and Pauer presented a basic theory of Groebner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a Groebner basis. In this paper, we will give a new criterion such that Insa and Pauer's criterion could be concluded as a special case and one could compute the Groebner basis more efficiently by this new criterion.

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arXiv2010

A New Proof for the Correctness of F5 (F5-Like) Algorithm

The famous F5 algorithm for computing Gröbner basis was presented by Faugère in 2002 without complete proofs for its correctness. The current authors have simplified the original F5 algorithm into an F5 algorithm in Buchberger's style (F5B algorithm), which is equivalent to original F5 algorithm and may deduce some F5-like versions. In this paper, the F5B algorithm is briefly revisited and a new complete proof for the correctness of F5B algorithm is proposed. This new proof is not limited to homogeneous systems and does not depend on the strategy of selecting critical pairs (i.e. the strategy deciding which critical pair is computed first) such that any strategy could be utilized in F5B (F5) algorithm. From this new proof, we find that the special reduction procedure (F5-reduction) is the key of F5 algorithm, so maintaining this special reduction, various variation algorithms become available. A natural variation of F5 algorithm, which transforms original F5 algorithm to a non-incremental algorithm, is presented and proved in this paper as well. This natural variation has been implemented over the Boolean ring. The two revised criteria in this natural variation are also able to reject almost all unnecessary computations and few polynomials reduce to 0 in most examples.

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