Related papers: Semigroup associated with a free polynomial

In this paper, we construct explicitely polynomial automorphisms of affine n-space for certain n. More precisely, we construct algebraic subgroups of the general polynomial group GA_n(k) where k is an arbitrary base ring of characteristic…

Let $R$ be a regular semi-local ring, essentially of finite type over an infinite perfect field of characteristic $p \ge 3$. We show that the cycle class map with modulus from an earlier work of the authors induces a pro-isomorphism between…

In this paper, we construct explicit families of polynomials $P \in \mathbb{F}_q[x_1,\dots,x_n]$ with large root sets which have restricted intersections with affine lines. We use these sets to make substantial progress on a number of…

Via counting over finite fields, we derive explicit formulas for the E-polynomials and Euler characteristics of GL(d)- and PGL(d)-character varieties of free groups. We prove a positivity property for these polynomials and relate them to…

The condition of plane polynomial curve to be a line in well-known Abhyankar-Moh Theorem is replaced by weaker ones. A criterion of embedded line is obtained from this strong theorem: Two polynomials can generate the entire polynomial ring…

We extend the free convolution of Brown measures of $R$-diagonal elements introduced by K\"{o}sters and Tikhomirov [Probab. Math. Statist. 38 (2018), no. 2, 359--384] to fractional powers. We then show how this fractional free convolution…

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our…

Recently several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from…

We investigate the density of square-free values of polynomials with large coefficients over the rational function field $\mathbb{F}_q[t]$. Some interesting questions answered as special cases of our results include the density of…

A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is…

We discuss the role of additive polynomials and $p$-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of…

This paper studies the representations of a non-negative polynomial $f$ on a non-compact semi-algebraic set $K$ modulo its critical ideal. Under the assumptions that the semi-algebraic set $K$ is regular and $f$ satisfies the boundary…

For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of…

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer…

This paper has two objectives: we first generalize the theory of Abhyankar-Moh to quasi-ordinary polynomials, then we use the notion of approximate roots and that of generalized Newton polygons in order to prove the embedding conjecture for…

Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…

For an algebraically closed field $K$ of characteristic zero and a non-singular matrix $A\in \mbox{GL}_n(K)$, a semi-invariant polynomial of $A$ is defined to be a polynomial $p(x)=p(x_1,\dots,x_n)$ with coefficients in $K$ such that…

Let $G$ be the group of $\mathbb R$--points of a semisimple algebraic group $\mathcal G$ defined over $\mathbb Q$. Assume that $G$ is connected and noncompact. We study Fourier coefficients of Poincar\' e series attached to matrix…

We prove Freudenburg's Freeness Conjecture: Let B be the polynomial ring in three variables over a field of characteristic zero, let D : B --> B be a nonzero locally nilpotent derivation, and let A = ker(D). Then B is a free A-module, and…