Related papers: Invariant varieties for polynomial dynamical syste…

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in $\cup_{\theta\in[0,\pi/2]}e^{i\theta}V$ is polynomially…

We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of the roots of a $C^{n-1,1}$-curve of monic polynomials of degree $n$ is locally…

The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives…

According to Medvedev and Scanlon, a polynomial $f(x)\in \bar{\mathbb Q}[x]$ of degree $d\geq 2$ is called disintegrated if it is not linearly conjugate to $x^d$ or $\pm C_d(x)$ (where $C_d(x)$ is the Chebyshev polynomial of degree $d$).…

We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In particular, for a complex number b we show that the derivative P_K^{(m,n)}(b,0)=d^m/da^m d^n/dx^n P_K(a,x)|(a, x)…

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 formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a…

Let K be an algebraically closed field of characteristic 0. Following Medvedev-Scanlon, a polynomial of degree d > 1 is said to be disintegrated if neither f nor -f is linearly conjugate to x^d or T_d(x) where T_d is the Chebyshev…

Let U(L) be the enveloping algebra of a finite dimensional Lie algebra L over a field k of characteristic zero, Z(U(L)) its center and Sz(U(L)) its semicenter. A sufficient condition is given in order for Sz(U(L)) to be a polynomial algebra…

This paper presents algebraic methods for the study of polynomial relative invariants, when the group G formed by the symmetries and relative symmetries is a compact Lie group. We deal with the case when the subgroup H of symmetries is…

In a previous paper by the author a universal ring of invariants for algebraic structures of a given type was constructed. This ring is a polynomial algebra that is generated by certain trace diagrams. It was shown that this ring admits the…

A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect…

Let p be a trigonometric polynomial, nonnegative on the unit circle $\mathbb{T}$. We say that a measure $\sigma$ on $\mathbb{T}$ belongs to the polynomial Szego class, if $d\sigma=sigma'_{ac}d\theta+d\sigma_s$, $\sigma_s$ is singular, and…

The symmetric group S_n acts on the skew polynomial ring A=k_{-1}[x_1,..., x_n] as permutations. For subgroups G of S_n we consider properties of the ring of invariants A^G. We show that A^G is always Artin-Schelter Gorenstein, and we…

We investigate deformations of skew group algebras that arise from a finite cyclic group acting on a polynomial ring in positive characteristic, where characteristic divides the order of the group. We allow deformations which deform both…

One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant…

We prove exact complexity dichotomies for two quantum invariants of closed oriented three-manifolds, with the categorical data fixed. For a modular category $\mathcal{C}$, computing the Reshetikhin--Turaev invariant $Z_{\mathcal{C}}(M)$…

We investigate the dynamical behaviour of a holomorphic map on a $f-$invariant subset $\mathcal{C}$ of $U,$ where $f:U \to \mathbb{C}^k.$ We study two cases: when $U$ is an open, connected and polynomially convex subset of $\mathbb{C}^k$…

We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…

We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…