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## Section: New Results

### Fundamental algorithms and structured polynomial systems

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

The so-called Berlekamp – Massey – Sakata algorithm computes a Gröbner basis of a 0-dimensional ideal of relations satisfied by an input table. It extends the Berlekamp – Massey algorithm to $n$-dimensional tables, for $n>1$.

In [1], we investigate this problem and design several algorithms for computing such a Gröbner basis of an ideal of relations using linear algebra techniques. The first one performs a lot of table queries and is analogous to a change of variables on the ideal of relations.

As each query to the table can be expensive, we design a second algorithm requiring fewer queries, in general. This FGLM -like algorithm allows us to compute the relations of the table by extracting a full rank submatrix of a multi-Hankel matrix (a multivariate generalization of Hankel matrices).

Under some additional assumptions, we make a third, adaptive, algorithm and reduce further the number of table queries. Then, we relate the number of queries of this third algorithm to the geometry of the final staircase and we show that it is essentially linear in the size of the output when the staircase is convex. As a direct application to this, we decode $n$-cyclic codes, a generalization in dimension $n$ of Reed Solomon codes.

We show that the multi-Hankel matrices are heavily structured when using the LEX ordering and that we can speed up the computations using fast algorithms for quasi-Hankel matrices. Finally, we design algorithms for computing the generating series of a linear recursive table.

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

In [22], we compare thoroughly the Berlekamp – Massey – Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multidimensional linear recurrent sequence.

Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.

#### Resultants and Discriminants for Bivariate Tensor-product Polynomials

Optimal resultant formulas have been systematically constructed mostly for unmixed polynomial systems, that is, systems of polynomials which all have the same support. However, such a condition is restrictive, since mixed systems of equations arise frequently in practical problems. In [16] we present a square, Koszul-type matrix expressing the resultant of arbitrary (mixed) bivariate tensor-product systems. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the entries of the matrix are simply coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. Moreover, for tensor-product systems with more than two (affine) variables, we prove an impossibility result: no universal degree-one formulas are possible, unless the system is unmixed. We also present applications of the new construction in the computation of discriminants and mixed discriminants as well as in solving systems of bivariate polynomials with tensor-product structure.

#### Sparse Rational Univariate Representation

In [15] we present explicit worst case degree and height bounds for the rational univariate representation of the isolated roots of polynomial systems based on mixed volume. We base our estimations on height bounds of resultants and we consider the case of 0-dimensional, positive dimensional, and parametric polynomial systems.

Multi-homogeneous polynomial systems arise in many applications. In [11], we provide bit complexity estimates for representing the solutions of these systems. These are the best currently known bounds. 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 is finite.

We do not only obtain bounds but an algorithm is also given for solving such systems. We give bit complexity estimates 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 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.

#### Improving Root Separation Bounds

Let $f$ be a polynomial (or polynomial system) with all simple roots. The root separation of $f$ is the minimum of the pair-wise distances between the complex roots. A root separation bound is a lower bound on the root separation. Finding a root separation bound is a fundamental problem, arising in numerous disciplines. In [7] we present two new root separation bounds: one univariate bound, and one multivariate bound. The new bounds improve on the old bounds in two ways: (1) The new bounds are usually significantly bigger (hence better) than the previous bounds. (2) The new bounds scale correctly, unlike the previous bounds. Crucially, the new bounds are not harder to compute than the previous bounds.

#### Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

The known algorithms approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time. They are, however, quite involved and require a high precision of computing when the degree of the input polynomial is large, which causes numerical stability problems. In [8] we observe that these difficulties do not appear at the initial stages of the algorithms, and in our present paper we extend one of these stages, analyze it, and avoid the cited problems, still achieving the solution within a nearly optimal complexity estimates, provided that some mild initial isolation of the roots of the input polynomial has been ensured. The resulting algorithms promise to be of some practical value for root-finding and can be extended to the problem of polynomial factorization, which is of interest on its own right. We conclude with outlining such an extension, which enables us to cover the cases of isolated multiple roots and root clusters.

#### 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, 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.

#### Sliding solutions of second-order differential equations with discontinuous right-hand side

In [2], we consider second-order ordinary differential equations with discontinuous right-hand side. We analyze the concept of solution of this kind of equations and determine analytical conditions that are satisfied by typical solutions. Moreover, the existence and uniqueness of solutions and sliding solutions are studied.

#### Sparse FGLM algorithms

Given a zero-dimensional ideal $I\subset 𝕂\left[{x}_{1},...,{x}_{n}\right]$ of degree $D$, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of [3] are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical FGLM algorithm. Combining all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over $40,000$. Such an implementation outperforms the Magma and Singular ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is $O\left(D\left({N}_{1}+nlogD\right)\right)$, where ${N}_{1}$ is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX Gröbner basis of $\sqrt{I}$ via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp–Massey–Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove its construction is free. With the asymptotic analysis of such sparsity, we are able to show for generic systems the complexity above becomes $O\left(\sqrt{6/n\pi }{D}^{2+\frac{n-1}{n}}\right)$.