Related papers: Polynomial compositions with large monodromy group…

Let $\varphi: {\mathbb P}^1 \longrightarrow {\mathbb P}^1$ be a rational map of degree greater than one defined over a number field $k$. For each prime ${\mathfrak p}$ of good reduction for $\varphi$, we let $\varphi_{\mathfrak p}$ denote…

The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more…

We compute the combinatorial Aomoto-Betti numbers $\beta_p(\mathcal{A})$ of a complex reflection arrangement. When $\mathcal{A}$ has rank at least $3$, we find that $\beta_p(\mathcal{A})\le 2$, for all primes $p$. Moreover,…

We classified finite orbits of monodromies of the Fuchsian system for five $2\times 2$ matrices. The explicit proof of this result is given. We have proposed a conjecture for a similar classification for $6$ or more $2\times 2$ matrices.…

In this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2n all of whose roots lie on the unit circle. In particular, we consider a set V_n of vectors in R^n…

We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra…

The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

A log generic hypersurface in $\mathbb{P}^n$ with respect to a birational modification of $\mathbb{P}^n$ is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic…

We prove that the arboreal Galois representations attached to certain unicritical polynomials have finite index in an infinite wreath product of cyclic groups, and we prove surjectivity for some small degree examples, including a new family…

A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A…

Let $f(x) \in K(x)$ be a quadratic polynomial where $K$ is a field of characteristic not equal to $2$. The associated arboreal Galois representation of the absolute Galois group of $K$ acts on a regular rooted binary tree. Boston and Jones…

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry.…

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes…

We introduce a new matroid (graph) invariant, the arboricity polynomial. Given a matroid, the arboricity polynomial enumerates the number of covers of the ground set by disjoint independent sets. We establish the polynomiality of the…

We realize several combinatorial Hopf algebras based on set compositions, plane trees and segmented compositions in terms of noncommutative polynomials in infinitely many variables. For each of them, we describe a trialgebra structure, an…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

A free-by-cyclic group can often be viewed as a mapping torus of a free group automorphism (monodromy) in multiple ways. What dynamical properties must these monodromies share, and to what extent are they invariant under quasi-isometries?…

Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms…

It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree…