arXiv subjects

Yao Sun

Publications and source records attributed to Yao Sun.

94 records · Page 6Linked to original sources

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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