Gröbner bases of ideals invariant under a commutative group: the non-modular case

We propose efficient algorithms to compute the Gröbner basis of an ideal I subset k[x₁,...,x_n] 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,...,x_n], 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.