Related papers: Rigidity of Lyapunov exponents for polynomials

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers, but that captures finer information. These "generalized Lyubeznik numbers" are defined as lengths of certain…

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\colon G\to \cc$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1 +\ldots +x_n )$ is decomposable;…

Let $K$ be a global field and $n > 1$ an integer. We show $n$ is composite if and only if there is an irreducible polynomial $f(x) \in K[x]$ of degree $n$ which is reducible $q$-adically for all the primes $q$ of $K$.

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…

It is well known that Sobolev embeddings can be improved in the presence of symmetries. In this article, we considere the situation in which given a domain $\Omega=\Omega_1 \times \Omega_2$ in $\mathbb{R}^N$ with a cylindrical symmetry, and…

Let ${\mathcal B}=\{b_i \}_{i=1}^\infty$ be a fixed sequence of pairwise distinct elements of a number field $k$. Given the integers $2\leq s \leq r$, assuming a quantitative version of Vojta's conjecture on the bounded degree algebraic…

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…

It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In…

In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on…

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to…

The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f=g(h) over finite fields…

In this paper, we classify, up to three possible exceptions, all monic, post-critically finite quadratic polynomials $f(x)\in \mathbb{Z}[x]$ with an iterate reducible module every prime, but all of whose iterates are irreducible over…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

A number of authors have proven explicit versions of Lehmer's conjecture for polynomials whose coefficients are all congruent to 1 modulo m. We prove a similar result for polynomials f(X) that are divisible in (Z/mZ)[X] by a polynomial of…

We investigate Drinfeld modular polynomials parametrizing $T$-isogenies between Drinfeld $\mathbb{F}_q[T]$-modules of rank $r\geq 2$. By providing an explicit classification of such isogenies, we derive explicit bounds on the $T$-degrees of…

This paper characterizes polynomials within molecules. We show that a geometrically finite polynomial of degree $d\geq2$ lies in a molecule if and only if all its critical points belong to maximal Fatou chains, and show that distinct…

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the…

Let F,G in C[x_1,...,x_n] be two polynomials in n variables x_1,...,x_n over the complex numbers field C. In this paper, we prove that if the degree of the Poisson bracket [F,G] is small enough then there are strict constraints for…

The following problem is studied: describe the triplets $(\Omega,g,\mu)$, $\mu=\rho\,dx$, where $g= (g^{ij}(x))$ is the (co)metric associated with the symmetric second order differential operator $L(f) = \frac{1}{\rho}\sum_{ij} \partial_i…

We show that for any positive integer $d$, there are families of switched linear systems---in fixed dimension and defined by two matrices only---that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function…