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The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite…
A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime $p$, reduce its coefficients mod $p$ and consider its action on the field $\mathbb{F}_p$. The questions of whether and…
A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains,…
Invariable generation is a topic that has predominantly been studied for finite groups. In 2014, Kantor, Lubotzky, and Shalev produced extensive tools for investigating invariable generation for infinite groups. Since their paper, various…
We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting…
Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the…
Our focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect…
In this paper, we consider sequences of polynomials that satisfy differential--difference recurrences. Our interest is motivated by the fact that polynomials satisfying such recurrences frequently appear as generating polynomials of integer…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a…
We study the set of algebraic objects known as vanishing polynomials (the set of polynomials that annihilate all elements of a ring) over general commutative rings with identity. These objects are of special interest due to their close…
In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…
We characterize the action of isotropic pseudodifferential operators on functions in terms of their action on Hermite functions. We show that an operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic pseudodifferential operator of…
It is known that the notion of a transitive subgroup of a permutation group $P$ extends naturally to the subsets of $P$. We study transitive subsets of the wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes the…
We develop a theory of formal multivariate polynomials over commutative rings by treating them as ring terms. Our main result is that two ring terms are s-equivalent (when expanded they yield the same standard polynomial) iff they are…
Representations of polynomial covariance type commutation relations by linear integral operators on $L_p$ over measures spaces are investigated. Necessary and sufficient conditions for integral operators to satisfy polynomial covariance…
Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual lambda-ring structure on these rings. From the representation-theoretical point…
The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ of distinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j = \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all different then…
Let $V=V_1 \otimes \cdots \otimes V_n$ be a vector space over an algebraically closed field $K$ of characteristic zero with $\dim(V_i)=d_i$. We study the ring of polynomial invariants $K[\operatorname{End}(V)^{\oplus…
This paper is a continuation of our papers \cite{EK1, EK2}. In \cite{EK2} we showed that for the root system $A_{n-1}$ one can obtain Macdonald's polynomials as weighted traces of intertwining operators between certain finite-dimensional…
Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan…