Gröbner Bases of Ideals Invariant under a Commutative Group: the Non-Modular Case
Résumé
We propose efficient algorithms to compute the Gröbner basis of an ideal $I\subset k[x_1,\dots,x_n]$ 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 $R=k[x_1,\dots,x_n]$, 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 \emph{abelian} versions of the classical algorithms $F_4$, $F_5$ or FGLM. Moreover, this new variant of $F_4/F_5$ 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 $F_4$. Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time.
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