Related papers: Arithmetic of Unicritical Polynomial Maps

We answer to a problem raised by recent work of Jelonek and Kurdyka: how can one detect by rational arcs the bifurcation locus of a polynomial map $\bR^n\to\bR^p$ in case $p>1$. We describe an effective estimation of the "nontrivial" part…

In this paper we show that for each $n$, there exists a simple graph whose reliability polynomial has at least $n$ inflection points.

An $n$-valued map is a set-valued continuous function $f$ such that $f(x)$ has cardinality $n$ for every $x$. Some $n$-valued maps will "split" into a union of $n$ single-valued maps. Characterizations of splittings has been a major theme…

In this article, we study the dynamics of the following family of rational maps with one parameter: \begin{equation*} f_\lambda(z)= z^n+\frac{\lambda^2}{z^n-\lambda}, \end{equation*} where $n\geq 3$ and $\lambda\in\mathbb{C}^*$. This family…

The number of distinct maps (pre-maps) with a single vertex and valence $d$ is computed for any value of $d$. The types of maps (pre-maps) that we consider depend on whether the underlaying graph (pre-graph) is signed or unsigned and…

We introduce a class of infinitely renormalizable, unicritical diffeomorphisms of the disk (with a non-degenerate "critical point"). In this class of dynamical systems, we show that under renormalization, maps eventually become…

We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm…

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number…

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines,…

We prove pointwise bounds for two-parameter families of Jacobi polynomials. Our bounds imply estimates for a class of functions arising from the spectral analysis of distinguished Laplacians and sub-Laplacians on the unit sphere in…

These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones.…

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice…

In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set…

We consider a non-polynomial cubic spline to develop the classes of methods for the numerical solution of singularly perturbed two-point boundary value problems. The proposed methods are second and fourth order accurate and applicable to…

We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z)…

In this note, we investigate the measure of singular sets and critical sets of real-valued solutions of elliptic equations in two dimensions. These singular sets and critical sets are finitely many points in the plane. Adapting the Carleman…

Real or complex polynomial mappings between affines spaces admitting a Lipschitz-trivial value are completely characterized.

This paper derives numerical bounds for and implements the splitting circle method for finding roots of a univariate polynomial in the presence of fixed precision.

We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the $abc$-conjecture. For unicritical polynomials of degree at least five, we require only the…