## Section: New Results

### Fundamental algorithms and structured polynomial systems

#### On the complexity of the F5 Gröbner basis algorithm

We study the complexity of Gröbner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system.

We give a bound on the number of polynomials of degree $d$ in a Gröbner basis computed by ${F}_{5}$ algorithm in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Gröbner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Gröbner bases with signatures that ${F}_{5}$ computes and use it to bound the complexity of the algorithm.

Our estimates show that the version of ${F}_{5}$ we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassen's multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.

#### On the complexity of computing Gröbner bases for weighted homogeneous systems

Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights $W=({w}_{1},...,{w}_{n})$, $W$-homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree $deg({X}_{1}^{{\alpha}_{1}}\cdots {X}_{n}^{{\alpha}_{n}})={\sum}_{i=1}^{n}{w}_{i}{\alpha}_{i}$. Gröbner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted homogeneous case. In [6] , we show that in this case, the complexity estimate for Algorithm ${F}_{5}$ (${\left(\genfrac{}{}{0pt}{}{n+{d}_{max}-1}{{d}_{max}}\right)}^{\omega}$ can be divided by a factor ${\left({\prod}_{i=1}{w}_{i}\right)}^{\omega}$). For zero-dimensional systems, the complexity of Algorithm FGLM $n{D}^{\omega}$ (where $D$ is the number of solutions of the system) can be divided by the same factor ${\left({\prod}_{i=1}{w}_{i}\right)}^{\omega}$. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of $W$-degree $({d}_{1},...,{d}_{n})$, these complexity estimates are polynomial in the weighted Bézout bound ${\prod}_{i=1}^{n}{d}_{i}/{\prod}_{i=1}^{n}{w}_{i}$. Furthermore, the maximum degree reached in a run of Algorithm ${F}_{5}$ is bounded by the weighted Macaulay bound ${\sum}_{i=1}^{n}({d}_{i}-{w}_{i})+{w}_{n}$, and this bound is sharp if we can order the weights so that ${w}_{n}=1$. For overdetermined semi-regular systems, estimates from the homogeneous case can be adapted to the weighted case. We provide some experimental results based on systems arising from a cryptography problem and from polynomial inversion problems. They show that taking advantage of the weighted homogeneous structure yields substantial speed-ups, and allows us to solve systems which were otherwise out of reach.

#### Linear Algebra for Computing Gröbner Bases of Linear Recursive Multidimensional Sequences

Sakata generalized the Berlekamp – Massey algorithm to $n$
dimensions in 1988. The Berlekamp – Massey – Sakata (bms )
algorithm can be used for finding a Gröbner basis of a 0-dimensional
ideal of relations verified by a table. We investigate this problem using
linear algebra techniques, with motivations such as accelerating change of
basis algorithms (fglm ) or improving their complexity.
In [12] , we first define and characterize
multidimensional linear
recursive sequences
for 0-dimensional ideals.
Under genericity assumptions, we propose a randomized preprocessing of the
table that corresponds to performing a linear change of coordinates on the
polynomials associated with the linear recurrences. This technique then
essentially reduces our problem to using the efficient 1-dimensional
Berlekamp – Massey (bm )
algorithm.
However, the number of probes to the table in this scheme may be elevated.
We thus consider the table in the *black-box* model: we assume
probing the
table is expensive and we minimize the number of probes to the table in our
complexity model.
We produce an fglm -like algorithm for finding the relations in the
table, which lets us use linear algebra techniques. Under some additional
assumptions, we make this algorithm adaptive and reduce further the number
of table probes.
This number can be estimated by counting the number of distinct elements in a
multi-Hankel matrix (a multivariate generalization of Hankel matrices); we can
relate this quantity with the *geometry* of the final staircase. Hence,
in favorable cases such as convex ones, the complexity is
essentially linear in
the size of the output. Finally, when using the lex ordering, we can
make use of fast structured linear algebra similarly to the Hankel
interpretation of Berlekamp – Massey.

#### Nearly optimal computations with structured matrices

In [9] we estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation. We supply them now as well as the Boolean complexity estimates for the important problems of multiplication of transposed Vandermonde matrix and its inverse by a vector. All known Boolean cost estimates for such problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer’s important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes, as well as the transposed Vandermonde matrices. It is known that the solution of Toeplitz, Hankel, Cauchy, Vandermonde, and transposed Vandermonde linear systems of equations is generally prone to numerical stability problems, and numerical problems arise even for multiplication of Cauchy, Vandermonde, and transposed Vandermonde matrices by a vector. Thus our FFT-based results on the Boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices, showing rather moderate growth of the complexity as the input size increases.