Related papers: p-Groups and the Polynomial Ring of Invariants Que…

If G is a finitely generated group, and A an algebraic group, then Hom(G,A) is a possibly reducible algebraic variety denoted by R_A(G). Here we define the profile function, P_d(R_A(G)), of the representation variety of G over A to be…

Let G be a reductive linear algebraic group over a field k. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Invariant theory tells that the ring of invariants A^G=H^0(G,A) is…

Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…

Let (N, G), where N is a normal subgroup of G<SL_n(C), be a pair of finite groups and V a finite-dimensional fundamental G-module. We study the G-invariants in the symmetric algebra S(V) by giving explicit formulas of the Poincar\'{e}…

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

For each nonnegative integer $i$, let $a_i$ be the number of $i$-subsets of $V(G)$ that induce an acyclic subgraph of a given graph $G$. We define $A(G,x) = \sum_{i \geq 0} a_i x^i$ (the generating function for $a_i$) to be the acyclic…

For some positive integer $m$, a real polynomial $P(x)=\sum\limits_{k=0}^ma_kx^k$ with $a_k\geqslant 0$ is called log-concave (resp. ultra log-concave) if $a_k^2\geqslant a_{k-1}a_{k+1}$ (resp. $a_k^2\geqslant…

Let V be a finite dimensional complex superspace and G a simple (or a ``close'' to simple) Lie superalgebra of matrix type, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of…

The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets…

Let $G$ be a complex simple Lie group, and $\mathfrak{g}$ its Lie algebra. It is well known that a finite-dimensional $G$-module $V$ carrying a nondegenerate invariant bilinear form gives rise to a Hamiltonian Poisson space with a quadratic…

Premet has conjectured that the nilpotent variety of any finite-dimensional restricted Lie algebra is an irreducible variety. In this paper, we prove this conjecture in the case of Hamiltonian Lie algebra. and show that its nilpotent…

A topological invariant of a polynomial map $p:X\to B$ from a complex surface containing a curve $C\subset X$ to a one-dimensional base is given by a rational second homology class in the compactification of the moduli space of genus $g$…

The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be…

In this article, we look at real split semisimple algebraic groups $\mathsf{G}$ with trivial center and faithful irreducible algebraic representations $\mathtt{R}$ of $\mathsf{G}$ on some vector space $\mathsf{V}$ which admit zero as a…

Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the field of fractions $\K$. Denote the ring of polynomials in $n$ variables over $\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by…

A univariate graph polynomial P(G;X) is weakly distinguishing if for almost all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that the clique polynomial and the independence polynomial are weakly distinguishing.…

Let $G$ be a group, $F$ a field, and $A$ a finite-dimensional central simple algebra over $F$ on which $G$ acts by $F$-algebra automorphisms. We study the ideals and subalgebras of $A$ which are preserved by the group action. Let $V$ be the…

A function $f:R\to R$, where $R$ is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative $p\in R[x]$. Based on this notion we introduce ring invariants which associate to $R$ the numbers…

We study the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. The ring of polyfunctions over a commutative ring $R$ with unit element is the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense…