Quantitative equidistribution for the solutions of systems of sparse polynomial equations

For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},..., x_n^{\pm1}] whose coefficients are not too big with respect to its directional resultants, we show that the solutions in the algebraic n-th dimensional complex torus of the system of equations f_1=\dots=f_n=0, are approximately equidistributed near the unit polycircle. This generalizes to the multivariate case a classical result due to Erdos and Turan on the distribution of the arguments of the roots of a univariate polynomial. We apply this result to bound the number of real roots of a system of Laurent polynomials, and to study the asymptotic distribution of the roots of systems of Laurent polynomials with integer coefficients, and of random systems of Laurent polynomials with complex coefficients.Comment: 29 pages, 2 figures. Revised version, accepted for publication in the American Journal of Mathematic

Extraction of cylinders and cones from minimal point sets

We propose new algebraic methods for extracting cylinders and cones from minimal point sets, including oriented points. More precisely, we are interested in computing efficiently cylinders through a set of three points, one of them being oriented, or through a set of five simple points. We are also interested in computing efficiently cones through a set of two oriented points, through a set of four points, one of them being oriented, or through a set of six points. For these different interpolation problems, we give optimal bounds on the number of solutions. Moreover, we describe algebraic methods targeted to solve these problems efficiently

Virtual Roots of a Real Polynomial and Fractional Derivatives

International audienceAfter the works of Gonzales-Vega, Lombardi, Mahé,\cite{Lomb1} and Coste, Lajous, Lombardi, Roy \cite{Lomb2}, we consider the virtual roots of a univariate polynomial $f$ with real coefficients. Using fractional derivatives, we associate to $f$ a bivariate polynomial $P_f(x,t)$ depending on the choice of an origin $a$, then two type of plan curves we call the FDcurve and stem of $f$. We show, in the generic case, how to locate the virtual roots of $f$ on the Budan table and on each of these curves. The paper is illustrated with examples and pictures computed with the computer algebra system Maple

On the cut-off phenomenon for the transitivity of randomly generated subgroups

38 pagesInternational audienceConsider $K\geq2$ independent copies of the random walk on the symmetric group $S_N$ starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time n\in\NN, let $G_n$ be the subgroup of $S_N$ generated by the $K$ positions of the chains. In the uniform transposition model, we prove that there is a cut-off phenomenon at time $N\ln(N)/(2K)$ for the non-existence of fixed point of $G_n$ and for the transitivity of $G_n$, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non-existence of a fixed point of $G_n$ appears at time of order $N^{1+\frac 2K}$ (at least for $K\geq3$), but there is no cut-off phenomenon. In the latter model, we recover a cut-off phenomenon for the non-existence of a fixed point at a time proportional to $N$ by allowing the number $K$ to be proportional to $\ln(N)$. The main tools of the proofs are spectral analysis and coupling techniques

Computing Monodromy via Continuation Methods on Random Riemann Surfaces

International audienceWe consider a Riemann surface $X$ defined by a polynomial $f(x,y)$ of degree $d$, whose coefficients are chosen randomly. Hence, we can suppose that $X$ is smooth, that the discriminant $\delta(x)$ of $f$ has $d(d-1)$ simple roots, $\Delta$, and that $\delta(0) \neq 0$ i.e. the corresponding fiber has $d$ distinct points $\{y_1, \ldots, y_d\}$. When we lift a loop 0 \in \gamma \subset \Ci - \Delta by a continuation method, we get $d$ paths in $X$ connecting $\{y_1, \ldots, y_d\}$, hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of $\Delta$. Multiplying families of ''neighbor'' transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups. Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.On considere une surface de Riemann dont l'equation f(x,y)=0 est un polynome dont les coefficients sont des variables aleatoires Gaussiennes standards, ainsi que sa projection p sur l'axe des x. Puis on etudie et calcule des generateurs du groupe de monodromie correspondant a p

Computation of some Hilbert functions related to Schubert Calculus

Document d'archive.International audienceWe describe the Hilbert functions of opposite big cells of Schubert varieties and their connections with combinatorics.On presente le calcul de fonctions de Hilbert de certaines varietes algebriques qui ont de remarquables proprites combinatoires: la fonction de Hilbert est un polynome

Intriguing Patterns in the Roots of the Derivatives of some Random Polynomials

Our observations show that the sets of real (respectively complex) roots of the derivatives of some classical families of random polynomials admit a rich variety of patterns looking like discretized curves. To bring out the shapes of the suggested curves, we introduce an original use of fractional derivatives. Then we set several conjectures and outline a strategy to explain the presented phenomena. This strategy is based on asymptotic geometric properties of the corresponding complex critical points sets.Nos observations montrent que les enembles de racines reelles (resp. complexes) des derivees de certaines familles de polynomes aleatoires ont une riche varietes motifs structures qui ressemblent a des courbes discretisees. Pour faire clairement apparaitre ces courbes, nous avons recours a une utilisation originale des derivees fractionnaires. Nous posons ensuite des conjectures et proposons une strategie pour expliquer les phenomenes observes. Celle-ci est basee sur des proprietes de symetrie asymptotique de l'ensemble des points critiques de nos polynomes quand leur degre tend vers l'infini

Computer Algebra Applied to a Solitary Waves Study

International audienceWe apply computer algebra techniques, such as algebraic computations of resultants and discriminants, certified drawing (with a guaranteed topology) of plane curves, to a problem in fluid dynamics: We investigate " capillary-gravity " solitary waves in shallow water, relying on the framework of the Serre-Green-Naghdi equations. So, we deal with two-dimensional surface waves, propagating in a shallow water of constant depth. By a differential elimination process, the study reduces to describing the solutions of an ordinary non linear first order differential equation, depending on two parameters. The paper is illustrated with examples and pictures computed with the computer algebra system Maple