Related papers: A geometrical construction for the polynomial inva…
A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals…
The space of n (ordered) points on the projective line, modulo automorphisms of the line, is one of the most important and classical examples of an invariant theory quotient, and is one of the first examples given in any course. Generators…
We consider differential rings of the form (K[x; y];D), where K is an algebraically closed field of characteristic zero and D : K[x; y] \to K[x; y] is a K-derivation. We study the Automorphism Group of such a ring and give criteria for…
We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…
The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In…
We study the ring generated over a field of characteristic 0 by noncommuting indeterminates {x_1,x_2,...,x_n} subject only to the relations x_i\sigma_k=\sigma_k x_i, for i,k=1,2,...,n, and their consequences, where \sigma_k…
This paper explores a natural action of the group $\mathrm{PGL}_2(\mathbb F_q)$ on the set of monic irreducible polynomials of degree at least two over a finite field $\mathbb F_q$. Our main results deal with the existence and number of…
We compute the number of orbits of pairs in a finitely generated torsion module (more generally, a module of bounded order) over a discrete valuation ring. The answer is found to be a polynomial in the cardinality of the residue field whose…
This is my master thesis, under the supervision of Professor Amiram Braun. We classify in these paper the Gorenstein invariant rings in the modular case, where the group that acts on the 3-variable polynomial ring is finite , and the Char…
The invariant theory of finite groups can connect the coding theory to the number theory. In this paper, under this conformity, we obtain the minimal generators of the rings of E-polynomials constructed from the groups related to…
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
This note considers a finite algebraic group $G$ acting on an affine variety $X$ by automorphisms. Results of Dufresne on polynomial separating algebras for linear representations of $G$ are extended to this situation. For that purpose, we…
Trigonometric invariants are defined for each Weyl group orbit on the root lattice. They are real and periodic on the coroot lattice. Their polynomial algebra is spanned by a basis which is calculated by means of an algorithm. The…
In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies,…
We find the generating set of SL-invariant polynomials in four qubits that are also invariant under permutations of the qubits. The set consists of four polynomials of degrees 2,6,8, and 12, for which we find an elegant expression in the…
We investigate the structure of ideals generated by binomials (polynomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many classical examples from algebraic geometry, and…
We study the algebras generated by restriction and induction operations on complex modules over dihedral groups. In the case where the orders of all dihedral groups involved are not divisible by four, we describe the relations, a basis, the…
We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of type $A_n$, $B_n$, and $D_n$, the coefficients of the characteristic polynomial of the reflection…
Nebe, Rains and Sloane studied the polynomial invariants for real and complex Clifford groups and they relate the invariants to the space of complete weight enumerators of certain self-dual codes. The purpose of this paper is to show that…
Positively graded algebras are fairly natural objects which are arduous to be studied. In this article we query quotients of non-standard graded polynomial rings with combinatorial and commutative algebra methods.