Symbolic Computation

The article considers linear functions of many (n) variables - multilinear polynomials (MP). The three-steps evaluation is presented that uses the minimal possible number of floating point operations for non-sparse MP at each step. The…

A distributed logic programming language with support for meta-programming and stream processing offers a variety of interesting research problems, such as: How can a versatile and stable data structure for the indexing of a large number of…

We are interested in the application of Machine Learning (ML) technology to improve mathematical software. It may seem that the probabilistic nature of ML tools would invalidate the exact results prized by such software, however, the…

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases…

For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations.…

Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an…

Let $\mathbf{K}$ be a field and $\phi$, $\mathbf{f} = (f_1, \ldots, f_s)$ in $\mathbf{K}[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\mathcal{S}_n$, the group of permutations of $\{1, \dots,…

Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F}…

Answering connectivity queries in semi-algebraic sets is a long-standing and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number…

We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…

We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of…

We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and…

In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to…

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several…

The paper deals with the analytic complexity of solutions to bivariate holonomic hypergeometric systems of the Horn type. We obtain estimates on the analytic complexity of Puiseux polynomial solutions to the hypergeometric systems defined…

We present a non-commutative algorithm for the multiplication of a 2x2-block-matrix by its transpose using 5 block products (3 recursive calls and 2 general products) over C or any finite field.We use geometric considerations on the space…

For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The…

Here we present some revised arguments to a randomized algorithm proposed by Sudan to find the polynomials of bounded degree agreeing on a dense fraction of a set of points in $\mathbb{F}^{2}$ for some field $\mathbb{F}$.

The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Groebner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced…

We consider the computation of syzygies of multivariate polynomials in a finite-dimensional setting: for a $\mathbb{K}[X_1,\dots,X_r]$-module $\mathcal{M}$ of finite dimension $D$ as a $\mathbb{K}$-vector space, and given elements…