Related papers: Higher Level Completeness for Permutation Polynomi…

We consider maps between commutative groups and their functional degrees. These degrees are defined based on a simple idea -- the functional degree should decrease if a discrete derivative is taken. We show that the maps of finite…

The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a…

Let R be a polynomial ring over k(u), where k is a field k and u is a finite family or inderterminates. The paper introduces the specialization of an arbitrary finitely generated R-module by the substitution of u to elements of k. This…

Let K be the quotient field of a complete local domain of dimension 2 with a separably closed residue field. Let G be a finite group of order not divisible by char(K). Then G is admissible over K if and only if its Sylow subgroups are…

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P(T) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a…

We present a new technique for computing permutation polynomials based on equivalence relations. The equivalence relations are defined by expanded normalization operations and new functions that map permutation polynomials (PPs) to other…

A permutation $\sigma=\sigma_1 \sigma_2 \cdots \sigma_n$ has a descent at $i$ if $\sigma_i>\sigma_{i+1}$. A descent $i$ is called a peak if $i>1$ and $i-1$ is not a descent. The size of the set of all permutations of $n$ with a given…

We show that if a field k contains sufficiently many elements(for instance, if k is infinite), and K is an algebraically closed field containing k, then every linear algebraic k-group over K is k-isomorphic to Aut(A\otimes_kK), where A is a…

A description of group automorphisms of all two-dimensional algebras, considered up to isomorphism, over any basic field is provided.

Given a polynomial \( H(x) \) over \(\mathbb{F}_{q^n}\), we study permutation polynomials of the form \( x + \gamma \mathrm{Tr}(H(x)) \) over \(\mathbb{F}_{q^n}\). Let \[P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is…

We estimate the number of possible types degree patterns of $k$-lacunary polynomials of degree $t < p$ which split completely modulo $p$. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with…

Let $f=ax+x^{r(q-1)+1}\in \mathbb{F}_{q^2}^*[x], r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of $\mathbb{F}_{q^2}.$

We define the operation of composing two hereditary classes of permutations using the standard composition of permutations as functions and we explore properties and structure of permutation classes considering this operation. We mostly…

In this paper we study the field of definition of abelian subvarieties $B\subset A_{\overline{K}}$ for an abelian variety $A$ over a field $K$ of characteristic $0$. We show that, provided that no isotypic component of $A_{\overline{K}}$ is…

For all finite fields of $q$ elements where $q\equiv1\pmod4$ we have constructed permutation polynomials which have order 2 as permutations, and have 3 terms, or 4 terms as polynomials. Explicit formulas for their coefficients are given in…

This paper classifies the contiguity relations for finite families of polynomials within the ($q$-)Askey scheme. The necessary and sufficient conditions for the existence of these contiguity relations are presented first. These conditions…

We develop the theory of difference algebraic groups in the case where we have finitely many pairwise commuting difference operators. We show that the defining ideal of a difference algebraic group is finitely generated as a difference…

A notion of Drinfeld polynomials is introduced for modules of two-parameter quantum affine algebras. Finite dimensional representations are then characterized by sets of $l$-tuples of pairs of Drinfeld polynomials with certain conditions.

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…