Non-commmutative elimination in ore algebras proves multivariate identities
Journal of Symbolic Computation
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Plural: a computer algebra system for noncommutative polynomial algebras
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Efficient algorithms for solving overdefined systems of multivariate polynomial equations
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Generalization of the F5 algorithm for calculating Gröbner bases for polynomial ideals
Programming and Computing Software
A new incremental algorithm for computing Groebner bases
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Journal of Symbolic Computation
F5C: A variant of Faugère's F5 algorithm with reduced Gröbner bases
Journal of Symbolic Computation
Signature-based algorithms to compute Gröbner bases
Proceedings of the 36th international symposium on Symbolic and algebraic computation
A generalized criterion for signature related Gröbner basis algorithms
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation
Modifying Faugère's F5 algorithm to ensure termination
ACM Communications in Computer Algebra
An efficient method for computing comprehensive Gröbner bases
Journal of Symbolic Computation
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Signature-based algorithms, including F5, F5C, G2V and GVW, are efficient algorithms for computing Gröbner bases in commutative polynomial rings. In this paper, we present a signature-based algorithm to compute Gröbner bases in solvable polynomial algebras which include usual commutative polynomial rings and some non-commutative polynomial rings like Weyl algebra. The generalized Rewritten Criterion (discussed in Sun and Wang, ISSAC 2011) is used to reject redundant computations. When this new algorithm uses the partial order implied by GVW, its termination is proved without special assumptions on computing orders of critical pairs. Data structures similar to F5 can be used to speed up this new algorithm, and Gröbner bases of syzygy modules of input polynomials can be obtained from the outputs easily. Experimental data show that most redundant computations can be avoided in this new algorithm.