Related papers: A geometrical construction for the polynomial inva…
By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, $d$, have a parameter space. The hyperbolic components…
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…
We show that the higher Pythagoras numbers for the polynomial ring are infinite $p_{2s}(K[x_1,x_2,\dots,x_n])=\infty$ provided that $K$ is a formally real field, $n\geq2$ and $s\geq 1$. This almost fully solves an old question \cite[Problem…
We study the number of ways of factoring elements in the complex reflection groups G(r,s,n) as products of reflections. We prove a result that compares factorization numbers in G(r,s,n) to those in the symmetric group on n letters, and we…
The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient…
We consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible…
We define a class of groups constructed from rings equipped with an involution. We show that under suitable conditions, these groups are either algebraic or arithmetic, including as special cases the orientation-preserving isometry group of…
We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3,…
This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear…
A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…
A polynomial of degree $n$ in two variables is shown to be uniquely determined by its Radon projections taken over $[n/2]+1$ parallel lines in each of the $(2[(n+1)/2]+1)$ equidistant directions along the unit circle.
We identify the algebra of regular functions on the space of quartic polynomials in three complex variables invariant under SL(3,C) with an algebra of meromorphic automorphic forms on the complex 6-ball. We also discuss the underlying…
Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field $\mathbb{F}_q$. We count the irreducible polynomials in $\mathbb{F}_q[x]$, of a given degree, which have the form $h(x)^{\mathrm{deg}\, f}\cdot…
We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…
We consider three bivariate polynomial invariants $P$, $A$, and $S$ for rooted trees, as well as a trivariate polynomial invariant $M$. These invariants are motivated by random destruction processes such as the random cutting model or site…
We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid $Inc(\mathbb{N})$ of strictly increasing functions.…
Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint…
An invariant of a model of genus one curve is a polynomial in the coefficients of the model that is stable under certain linear transformations. The classical example of an invariant is the discriminant, which characterizes the singularity…
We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm…
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\…