## Section: New Results

### Zero dimensional Solve

Let $I\subset \mathbb{K}[{x}_{1},...,{x}_{n}]$ be a 0-dimensional ideal
of degree $D$ where $\mathbb{K}$ 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 $\mathbb{K}[{x}_{1},...,{x}_{n}]/I$. 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.