Related papers: Polynomial compositions with large monodromy group…

Let $f$ be a polynomial with integer coefficients whose degree is at least 2. We consider the problem of covering the orbit $\operatorname{Orb}_f(t)=\{t,f(t),f(f(t)),\cdots\}$, where $t$ is an integer, using arithmetic progressions each of…

Let $f\colon X \dashrightarrow X$ be a birational transformation of a projective manifold $X$ whose Kodaira dimension $\kappa(X)$ is non-negative. We show that, if there exist a meromorphic fibration $\pi \colon X\dashrightarrow B$ and a…

Honda and Tate showed that the isogeny classes of abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$ are classified in terms of $q$-Weil polynomials of degree $2g$, that is, monic integer polynomials whose set of complex…

Let $f(x) \in \mathbb{F}_p[x]$, and define the orbit of $x\in \mathbb{F}_p$ under the iteration of $f$ to be the set \[ \mathcal{O}(x):=\{x,f(x),(f\circ f)(x),(f\circ f\circ f)(x),\dots\}. \] An orbit is a $k$-cycle if it is periodic of…

This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear…

We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of $f(z)=z^2-1$ on associated coverings of the Riemann sphere. We then derive some algebraic properties of the group, and…

We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such…

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential $\left\langle…

We study representations of groups by "affine" automorphisms of compact, convex spaces, with special focus on "irreducible" representations: equivalently "minimal" actions. When the group in question is PSL(2,R), we exhibit a one-one…

The investigation and classification of non-unique factorization phenomena has attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P.A. Garc\'ia-S\'anchez, together with several co-authors, derived…

We introduce a graded Hopf algebra based on the set of parking functions (hence of dimension (n+1)^{n-1} in degree n). This algebra can be embedded into a noncommutative polynomial algebra in infinitely many variables. We determine its…

Orbits of automorphism groups of partially ordered sets are not necessarily congruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that…

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of…

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there…

Let $\mathbb{F}$ denote an algebraically closed field with characteristic $0$, and let $q$ denote a nonzero scalar in $\mathbb{F}$ that is not a root of unity. Let $\mathbb{Z}_4$ denote the cyclic group of order $4$. Let $\square_q$ denote…

A combinatorial construction is used to analyze the properties of polyhedral products and generalized moment-angle complexes with respect to certain operations on CW pairs including exponentiation. This allows for the construction of…

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…

For a class of polynomials $f \in \mathbb{Z}[X]$, which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set…

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We…

We introduce and study Hopf monads on autonomous categories (i.e., monoidal categories with duals). Hopf monads generalize Hopf algebras to a non-braided (and non-linear) setting. Indeed, any monoidal adjunction between autonomous…