## Section: New Results

### Fundamental algorithms and structured polynomial systems

#### Towards Mixed Gröbner Basis Algorithms: the Multihomogeneous and Sparse Case

One of the biggest open problems in computational algebra is the design of efficient algorithms for Gröbner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed polynomial systems, that is systems with polynomials having the same support, using the approach of Faugère, Spaenlehauer, and Svartz [ISSAC'14]. In [15] we present two algorithms for sparse Gröbner bases computations for mixed systems. The first one computes with mixed sparse systems and exploits the supports of the polynomials. Under regularity assumptions, it performs no reductions to zero. For mixed, square, and 0-dimensional multihomogeneous polynomial systems, we present a dedicated, and potentially more efficient, algorithm that exploits different algebraic properties that performs no reduction to zero. We give an explicit bound for the maximal degree appearing in the computations.

#### Bilinear Systems with Two Supports: Koszul Resultant Matrices, Eigenvalues, and Eigenvectors

A fundamental problem in computational algebraic geometry is the computation
of the resultant. A central question is when and how to compute it
as the determinant of a matrix
whose elements are the coefficients of the input polynomials up-to sign.
This problem is well understood for unmixed
multihomogeneous systems, that is for systems consisting of multihomogeneous
polynomials with the same support.
However, little is known for mixed systems,
that is for systems consisting of polynomials with different supports.
In [14] we consider the computation of the
multihomogeneous resultant of
bilinear systems involving two different supports. We present a
constructive approach that expresses the resultant as the exact
determinant of a *Koszul resultant matrix*, that is a matrix
constructed from maps in the Koszul complex.
We exploit the resultant matrix to propose an algorithm to solve
such systems. In the process we extend the classical eigenvalues
and eigenvectors criterion to a more general setting. Our extension
of the eigenvalues criterion applies to a general class of matrices,
including the Sylvester-type and the Koszul-type ones.

#### A Polynomial-Division-Based Algorithm for Computing Linear Recurrence Relations

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence.

Several algorithms solve this problem. The so-called Berlekamp–Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process.

In [16], we propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations.

A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial.

Finally, we give a partial solution to the transformation of this algorithm into an adaptive one.

As an addition from the paper published at the ISSAC conferance, in [24], we give an adaptive variant of this algorithm taking into account the shape of the final Gröbner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gröbner basis.

All these algorithms have been implemented in Maple and we report on our comparisons.

#### In-depth comparison of the Berlekamp–Massey–Sakata and the Scalar-FGLM algorithms: the adaptive variants

The Berlekamp–Massey–Sakata algorithm and the Scalar-FGLM algorithm both compute the ideal of relations of a multidimensional linear recurrent sequence.

Whenever quering a single sequence element is prohibitive, the bottleneck of these algorithms becomes the computation of all the needed sequence terms. As such, having adaptive variants of these algorithms, reducing the number of sequence queries, becomes mandatory.

A native adaptive variant of the Scalar-FGLM algorithm was presented by its authors, the so-called Adaptive Scalar-FGLM algorithm.

In [25], our first contribution is to make the Berlekamp–Massey–Sakata algorithm more efficient by making it adaptive to avoid some useless relation testings. This variant allows us to divide by four in dimension 2 and by seven in dimension 3 the number of basic operations performed on some sequence family.

Then, we compare the two adaptive algorithms. We show that their behaviors differ in a way that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other. We detail precisely the differences and the similarities of both algorithms and conclude that in general the Adaptive Scalar-FGLM algorithm needs fewer queries and performs fewer basic operations than the Adaptive Berlekamp–Massey–Sakata algorithm.

We also show that these variants are always more efficient than the original algorithms.

#### Bit complexity for multi-homogeneous polynomial system solving Application to polynomial minimization

Multi-homogeneous polynomial systems arise in many applications. In [10] we provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set if finite. The algorithm is probabilistic and a probability analysis is provided. Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.