GALLIGO , André

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...

We describe the Hilbert functions of opposite big cells of Schubert varieties and their connections with combinatorics.

We first recall the main features of Fractional calculus. In the expression of fractional derivatives of a real polynomial $f(x)$, we view the order of differentiation $q$ as a new indeterminate; then we define a new bivariate polynomial $P_f(x,q)$. For $0 \leq q \leq 1$, $P_f(x,q)$ defines an homotopy between the polynomials $f(x)$ and...

The Budan table of f collects the signs of the iterated derivative of f. We revisit the classical Budan-Fourier theorem for a univariate real polynomial f and establish a new connexity property of its Budan table. We use this property to characterize the virtual roots of f, (introduced by Gonzales-Vega, Lombardi, Mahe in 1998); they are...

Given a degree n univariate polynomial f(x), the Budan-Fourier function Vf (x) counts the sign changes in the sequence of derivatives of f evaluated at x. The values at which this function jumps are called the virtual roots of f, these include the real roots of f and any multiple root of its derivatives. This concept was introduced (by an...

In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger...

After 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...

International audience

Given two curves in the projective space, either implicitly or by a parameterization, we want to check if they intersect. For that purpose, we present and further develop generalized resultant techniques. Our aim is to provide a closed formula in the inputs which vanishes if and only if the two curves intersect. This could be useful in Computer...

We 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 $\{y_1, \ldots, y_d\}$....

Consider $K\geq2$ 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...

The intersection curve of two parameterized surfaces is characterized by 3 equations $F_i(s,t) = G_i(u,v)$, $i = 1, 2, 3$ of 4 variables. So, it is the image of a curve in four dimensional space. We provide a method to draw such curve with a guaranteed topology.

We consider a univariate polynomial f with real coefficients having a high degree $N$ but a rather small number $d+1$ of monomials, with $d\ll N$. Such a sparse polynomial has a number of real root smaller or equal to $d$. Our target is to find for each real root of $f$ an interval isolating this root from the others. The usual subdivision...

International audience

A new algebraic method for extracting tori from a minimal point set, made of two oriented points and a simple point, is proposed. We prove a degree bound on the number of such tori; this bound is reached on examples, even when we restrict to smooth tori. Our method is based on pre-computed closed formulae well suited for numerical computations...

In this paper we introduce an intermediate representation of surfaces that we call semi-implicit. We give a general definition in the language of projective complex algebraic geometry, and we begin its systematic study with an effective view point. Our last section will apply this representation to investigate the intersection of two bi-cubic...

We consider mixed polynomials P (z, ¯ z) of the single complex variable z with complex (or real) coefficients, of degree n in z and m in ¯ z. This data is equivalent to a pair of real bivariate polynomials f (x, y) and g(x, y) obtained by separating real and imaginary parts of P. However specifying the degrees, here we focus on the case where m...

International audience