Related papers: Local permutation polynomials and their companions

For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to…

In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite…

We study linearizability properties of finite subgroups of the Cremona group ${\mathrm{Cr}}_n(k)$ in the case where $k$ is a global field, with the focus on the local-global principle. For every global field $k$ of characteristic different…

The paper studies the question of existence of polynomials with given roots over associative non-commutative rings with identity. It is shown that in the case of an associative division ring for arbitrary n elements of this ring there…

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of…

We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials are related to a new algebraic characterization of abnormal subriemannian geodesics in stratified…

Given a finite field $\F_q$ and $n\in \N^*$, one could try to compute all polynomial endomorphisms $\F_q^n\lp \F_q^n$ up to a certain degree with a specific property. We consider the case $n=3$. If the degree is low (like 2,3, or 4) and the…

The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are $\gamma$-positive polynomials and can be interpreted as $h$-polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of…

Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize all Galois…

This manuscript defines a new family of invariants, analogous to the Lyubeznik numbers, associated to any local ring whose residue field has prime characteristic. In particular, as their nomenclature suggests, these "Lyubeznik numbers in…

We prove a low characteristic counterpart to the main result in (Peluse, 2019), establishing power saving bounds for the polynomial Szemer\'{e}di theorem for certain families of polynomials. Namely, we show that if $P_1, \dots, P_m \in…

Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs $(f,\alpha)$, where $f$ is a polynomial over a number field $K$ and $\alpha\in K$, such that the dynamical Galois group of the pair $(f,\alpha)$ is abelian. In…

Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension. Let $\pi_L$ be a uniformizer for $L$ and let $f(X)$ be the minimum polynomial for $\pi_L$ over $K$. Suppose…

R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials over $\mathbb{F}_{q^3}$ in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed…

This paper provides a realization of all classical and most exceptional finite groups of Lie type as Galois groups over function fields over F_q and derives explicit additive polynomials for the extensions. Our unified approach is based on…

We give necessary and sufficient conditions for a polynomial of the form x^r*(1+x^v+x^(2v)+...+x^(kv))^t to permute the elements of the finite field GF(q). Our results yield especially simple criteria in case (q-1)/gcd(q-1,v) is a small…

We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this…

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…

We give algorithms to compute isomorphism classes of ordinary abelian varieties defined over a finite field $\mathbb{F}_q$ whose characteristic polynomial (of Frobenius) is square-free and of abelian varieties defined over the prime field…