Section: New Results

Zero dimensional Solve

Let $I\subset 𝕂[x_1,...,x_n]$ be a 0-dimensional ideal of degree $D$ where $𝕂$ is a field. It is well-known that obtaining efficient algorithms for change of ordering of Gröbner bases of $I$ is crucial in polynomial system solving. Through the algorithm FGLM, this task is classically tackled by linear algebra operations in $𝕂[x_1,...,x_n]/I$. With recent progress on Gröbner bases computations, this step turns out to be the bottleneck of the whole solving process.

In [38] , we present an efficient algorithm that takes advantage of the sparsity structure of multiplication matrices appearing during the change of ordering. This sparsity structure arises even when the input polynomial system defining $I$ is dense. As a by-product, we obtain an implementation which is able to manipulate 0-dimensional ideals over a prime field of degree greater than 30000. It outperforms the Magma/Singular/FGb implementations of FGLM.

In [38] , we investigate the particular but important shape position case. The obtained algorithm performs the change of ordering within a complexity $O\left(D(N_1+nlog\left(D\right))\right)$, where $N_1$ is the number of nonzero entries of a multiplication matrix (the density of the matrix). This almost matches the complexity of computing the minimal polynomial of one multiplication matrix. Then, we address the general case and give corresponding complexity results. Our algorithm is dynamic in the sense that it selects automatically which strategy to use depending on the input. Its key ingredients are the Wiedemann algorithm to handle 1-dimensional linear recurrence (for the shape position case), and the Berlekamp–Massey–Sakata algorithm from Coding Theory to handle multi-dimensional linearly recurring sequences in the general case.