arXiv subjects
Publications and source records attributed to Mohamed Chaouki Tarchi.
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where traditional methods might fall short.
We consider real univariate degree $d$ real-rooted polynomials with non-vanishing coefficients. Descartes' rule of signs implies that such a polynomial has $\tilde{c}$ positive and $\tilde{p}$ negative roots counted with multiplicity, where $\tilde{c}$ and $\tilde{p}$ are the numbers of sign changes and sign preservations in the sequence of its coefficients, $\tilde{c}+\tilde{p}=d$. For $d=6$, we give the exhaustive answer to the question: When the moduli of all $6$ roots are distinct and arranged on the real positive half-axis, in which positions can the moduli of the negative roots be depending on the signs of the coefficients?
The {\em sign pattern} defined by the real polynomial $Q:=Σ_{j=0}^da_jx^j$, $a_j\neq 0$, is the string $σ(Q):=({\rm sgn(}a_d{\rm )},\ldots ,{\rm sgn(}a_0{\rm )})$. The quantities $pos$ and $neg$ of positive and negative roots of $Q$ satisfy Descartes' rule of signs. A couple $(σ_0,(pos,neg))$, where $σ_0$ is a sign pattern of length $d+1$, is {\em realizable} if there exists a polynomial $Q$ with $pos$ positive and $neg$ negative simple roots, with $(d-pos-neg)/2$ complex conjugate pairs and with $σ(Q)=σ_0$. We present a series of couples (sign pattern, pair $(pos,neg)$) depending on two integer parameters and with $pos\geq 1$, $neg\geq 1$, which is not realizable. For $d=9$, we give the exhaustive list of realizable couples with two sign changes in the sign pattern.
We consider the set of monic degree $d$ real univariate polynomials $Q_d=x^d+\sum_{j=0}^{d-1}a_jx^j$ and its {\em hyperbolicity domain} $Π_d$, i.e. the subset of values of the coefficients $a_j$ for which the polynomial $Q_d$ has all roots real. The subset $E_d\subset Π_d$ is the one on which a modulus of a negative root of $Q_d$ is equal to a positive root of $Q_d$. At a point, where $Q_d$ has $d$ distinct roots with exactly $s$ ($1\leq s\leq [d/2]$) equalities between positive roots and moduli of negative roots, the set $E_d$ is locally the transversal intersection of $s$ smooth hypersurfaces. At a point, where $Q_d$ has two double opposite roots and no other equalities between moduli of roots, the set $E_d$ is locally the direct product of $\mathbb{R}^{d-3}$ and a hypersurface in $\mathbb{R}^3$ having a Whitney umbrella singularity. For $d\leq 4$, we draw pictures of the sets $Π_d$ and~$E_d$.