Symbolic solution polynomial equation systems with symmetry
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
NTRU: A Ring-Based Public Key Cryptosystem
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation)
Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation)
Analytic Combinatorics
Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Gröbner bases of symmetric ideals
Journal of Symbolic Computation
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We propose efficient algorithms to compute the Gröbner basis of an ideal I subset k[x1,...,xn] globally invariant under the action of a commutative matrix group G, in the non-modular case (where char(k) doesn't divide |G|). The idea is to simultaneously diagonalize the matrices in G, and apply a linear change of variables on I corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on I are diagonal. This action induces a grading on the ring R=k[x1,...,xn], compatible with the degree, indexed by a group related to G, that we call G-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into |G| submatrices of roughly the same size. In the same way, we are able to split the canonical basis of R/I (the staircase) if I is a zero-dimensional ideal. Therefore, we derive abelian versions of the classical algorithms F4, F5 or FGLM. Moreover, this new variant of F4/ F5 allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of F4. Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.