Related papers: Counting absolutely cuspidals for quivers

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…

The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f(x) = g(h(x)) in F[x] over a field F is well understood in…

We study representations of the Loop Kac-Moody Lie algebra g \otimes A, where g is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

Closed form expressions for a multivector exponential and logarithm are presented in real Clifford geometric algebras Cl(p,q)when n=p+q=1 (complex and hyperbolic numbers) and n=2 (Hamilton, split and conectorine quaternions). Starting from…

The $q$-binomial coefficients are q-analogues of the binomial coefficients, counting the number of $k$-dimensional subspaces in the $n$-dimensional vector space $\mathbb{F}^n_q$ over $\mathbb{F}_{q}$. In this paper, we define a Euclidean…

We provide a simple and new induction based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and…

We study symmetries enjoyed by the polynomials enumerating non-degenerate flags in finite vector spaces, equipped with a non-degenerate alternating bilinear, hermitian or quadratic form. To this end we introduce Igusa-type rational…

We study a quantum version of the Kazhdan-Lusztig functor. Namely, we prove that there exists a fully faithfull exact tensor functor from the category of finite dimensional representations of the quantum affine algebra Uq(sl(n)) (with…

In this note we give a theoretical support by means of quotient polynomial rings for the computation formulas of the dimension of abelian codes.

An explicit invariant-theoretic description of the moduli space $\mathcal{M}_3^1$ of degree-three rational maps on $\mathbb{P}^1$ is developed. A cubic map $\phi$ is represented, up to conjugation, by the pair of binary forms $(f, g) \in…

Let $V$ be a vector space over a finite field $k=\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac {|\sum _{v\in V}\psi (P(V))|}{q^n}$$ where $\psi :k\to \mathbb{C} ^\star$ is a…

In recent series of works, by translating properties of multi-centered supersymmetric black holes into the language of quiver representations, we proposed a formula that expresses the Hodge numbers of the moduli space of semi-stable…

This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating…

In this paper we compute the sum of the $k$-th powers over any finite commutative unital rings, thus generalizing known results for finite fields, the rings of integers modulo $n$ or the ring of Gaussian integers modulo $n$. As an…

The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=\mathbf{F}[x_n,...,x_1]$, a polyonomial ring over the finite field $\mathbf{F}:=\mathbf{Z}_q$ of $q$…

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are…

Let $l$ be a finite field of cardinality $q$ and let $n$ be in $\mathbb{Z}_{\geq 1}$. Let $f_1,\ldots,f_n \in l[x_1,\ldots,x_n]$ not all constant and consider the evaluation map $f=(f_1,\ldots,f_n) \colon l^n \to l^n$. Set…

Let $\Aa_t$ be the directed quiver of type $\Aa$ with $t$ vertices. For each dimension vector $d$ there is a dense orbit in the corresponding representation space. The principal aim of this note is to use just rank conditions to define the…

This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over…