Face au constat d’une heterogeneite grandissante des savoir-faire et connaissances en informatique des etudiants a l’arrivee en premiere annee, et le risque de son exacerbation dans le contexte du « nouveau bac », nous avons voulu experimenter une approche pedagogique, qui permette une gestion de cette heterogeneite tout en respectant les contraintes d’un emploi du temps homogene et un cout constant. Les actions menees s’articulent autour de 4 poles : la constitution de groupes de niveau, avec une attention particuliere portee sur les 2 niveaux extremes (renforcement et avance/en autonomie), la mise en place de QCMs reguliers, l’utilisation ponctuelle de l’Apprentissage Par Probleme (APP), et un auto-positionnement. L’experimentation est encore en cours, mais deja de premiers elements permettent d’ouvrir les echanges.

A sum of affine powers is an expression of the form [f(x 1 ,...,x n ) = ∑ i =1 s α i l i (x 1 ,...,x n ) e i ] where l i is an affine form. We propose polynomial time black-box algorithms that find the decomposition with the smallest value of s for an input polynomial f . Our algorithms work in situations where s is small enough compared to the number of variables or to the exponents e i . Although quite simple, this model is a generalization of Waring decomposition. This paper extends previous work on Waring decomposition as well as our work on univariate sums of affine powers (ISSAC'17).

We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by $F$. In particular, we give simple criteria ensuring that the dimension of the span of $F$ is at least $c.|F|$ for some absolute constant $c<1$. We also propose conjectures implying the linear independence of the elements of $F$. These conjectures are known to be true for the field of real numbers, but not for the field of complex numbers.

In this paper we study sums of powers of affine functions in (mostly) one variable. Although quite simple, this model is a generalization of two well-studied models: Waring decomposition and sparsest shift. For these three models there are natural extensions to several variables, but this paper is mostly focused on univariate polynomials. We present structural results which compare the expressive power of the three models; and we propose algorithms that find the smallest decomposition of f in the first model (sums of affine powers) for an input polynomial f given in dense representation. We also begin a study of the multivariate case. This work could be extended in several directions. In particular, just as for Sparsest Shift and Waring decomposition, one could consider extensions to "supersparse" polynomials and attempt a fuller study of the multi-variate case. We also point out that the basic univariate problem studied in the present paper is far from completely solved: our algorithms all rely on some assumptions for the exponents in an optimal decomposition, and some algorithms also rely on a distinctness assumption for the shifts. It would be very interesting to weaken these assumptions, or even to remove them entirely. Another related and poorly understood issue is that of the bit size of the constants appearing in an optimal decomposition: is it always polynomially related to the bit size of the input polynomial given in dense representation?

Consider a group of stanchions linked together in a waiting line. In order to paint both sides of every stanchion you will need to lift your paintbrush as many times as the number of faces of the corresponding plane graph. As a lazy graph theorist you want to twist the strips between stanchions in a Möbius fashion such that you do not need to lift up your paintbrush. We call such a twist a MSS and we investigate the space of all MSSs of a planar graph. Our main results are that all the MSSs are connected by a series of two elementary operations, and that the space of MSSs does not depend on the planar embedding of the graph.

The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives? We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the trace method, recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.A slightly shorter version of this paper was presented at STACS'17. In this new version we have corrected a typo in Section 4.1, and added a reference to Shitov's work on tensor rank.

We consider the problem of representing a univariate polynomial f(x) as a sum of powers of low degree polynomials. We prove a lower bound of Ω( √(d/t)) for writing an explicit univariate degree-d polynomial f(x) as a sum of powers of degree-t polynomials.

This is the nal report of an internship in algebraic complexity. First, we give an introduction to algebraic complexity and we give some motivations for the study of the main model. Then we present two di erent tools we studied during this internship and use them to establish some lower bounds on this model. We nally discuss whether those bounds could be improved or not.

In this note, we prove that all cop-win graphs $G$ in the game in which the robber and the cop move at different speeds $s$ and $s'$ with $s'0$, this establishes a new---game-theoretical---characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between $\delta$ and $s$ is linear for any $s'