Related papers: Real Polynomial Diffeomorphisms with Maximal Entro…

In this short note we provide the example with the lowest degree known so far of a non-injective local polynomial diffeomorphism $F=(p,q):\mathbb{R}^2 \to \mathbb{R}^2$. In our example $p$ has degree $10$ and $q$ has degree $15$, rather…

Let $F:\mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$ be a $\mathbb{C}$-algebra endomorphism that has an invertible Jacobian. We bring two ideas concerning the Jacobian Conjecture: First, we conjecture that for all $n$, the…

Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other…

In this paper together with the preceding Part I \cite{CHR}, we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to \cite{CHR} the treatment of the mixed…

We determine and classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradicals are isomorphic to dual Radford algebras of dimension $4p$ for a prime $p>5$. In particular, we…

We give an estimate for the Hausdorff dimension of the bifurcation locus of a family of endomorphisms of $\mathbb{P}^k (\mathbb{C})$. This dimension is maximal near isolated Latt\`es examples.

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that…

Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…

We describe results on the dynamics of polynomial diffeomorphisms of ${\bf C^2}$ and draw connections with the dynamics of polynomial maps of ${\bf C}$ and the dynamics of polynomial diffeomorphisms of ${\bf R^2}$ such as the H\'enon…

We prove that the jacobian of a hyperelliptic curve y^2=f(x) is absolutely simple if deg(f)=q+1 where q is a power prime congruent to 5 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois…

We derive a sufficient condition for topological horseshoe and uniform hyperbolicity of a 4-dimensional symplectic map, which is introduced by coupling the two 2-dimensional H\'enon maps via linear terms. The coupled H\'enon map thus…

In this paper, we show that for a given degenerate bivector $\pi= y^n\partial_x \wedge \partial_y$ with $n>1$, the classical Poisson cohomology group and the logarithmic Poisson cohomology group along the ideal…

Given an algebraic family $f\colon\Lambda\times \mathbb{A}^2 \to \Lambda\times \mathbb{A}^2$ of plane polynomial automorphisms of H\'enon type parameterized by a quasi-projective curve, defined over a number field $\mathbb{K},$ we…

Let $K$ be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for $K$ are not…

The direct parametrisation method for invariant manifolds is adjusted to consider a varying parameter. More specifically, the case of systems experiencing a Hopf bifurcation in the parameter range of interest are investigated, and the…

Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of…

We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each {\epsilon} > 0, any {\epsilon}-approximated formulation of P in the original space R^n requires a number of inequalities that…

This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in…

We prove the existence of limits of real-analytic Laplace eigenvalue branches for real-analytic families of metrics that degenerate along a compact hypersurface.

We give an example of a family of endomorphisms of $\mathbb P^2 (\mathbb C)$ whose Julia set depends continuously on the parameter and whose bifurcation locus has non empty interior.