Graded-division algebras and Galois extensions

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras.On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field $\mathbb{F}$ is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of $\mathbb{F}$ and the isomorphism class of a G-Galois extension of $\mathbb{F}$.This connection is used to classify the simple G-Galois extensions of $\mathbb{F}$ in terms of a Galois field extension $\mathbb{L}/\mathbb{F}$ with Galois group isomorphic to a quotient $G/K$ and an element in the quotient ${\mathrm{\text{Z}}}^2(K,{\mathbb{L}}^{\times })/{\mathrm{\text{B}}}^2(K,{\mathbb{F}}^{\times })$ subject to certain conditions. Non-simple G-Galois extensions are induced from simple T-Galois extensions for a subgroup T of G. We also classify finite-dimensional G-graded-division algebras and, as an application, finite G-graded-division rings.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Classification of cyclic codes over a non-Galois chain ring Zp[u]/〈u3〉

We explicitly determine generators of cyclic codes over a non-Galois finite chain ring ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$, where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$ and four types of non-principal ideals of ${\mathbb{Z}}_p[u]/〈u^3〉$, which are associated with cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$. We then obtain a mass formula for cyclic codes over ${\mathbb{Z}}_p[u]/〈u^3〉$ of length $p^k$.     

The Hilbert series and a-invariant of circle invariants

Let V be a finite-dimensional representation of the complex circle ${\mathbb{C}}^{\times }$ determined by a weight vector $\mathit{a}\in {\mathbb{Z}}^n$. We study the Hilbert series ${\mathrm{Hilb}}_{\mathit{a}}(t)$ of the graded algebra $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ of polynomial ${\mathbb{C}}^{\times }$-invariants in terms of the weight vector a of the ${\mathbb{C}}^{\times }$-action. In particular, we give explicit formulas for ${\mathrm{Hilb}}_{\mathit{a}}(t)$ as well as the first four coefficients of the Laurent expansion of ${\mathrm{Hilb}}_{\mathit{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.     

Non-integrated defect relation for meromorphic mappings on Kähler manifolds intersecting non-subdegenerate families of hyperplanes

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m(R_0)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\infty$). Our purpose of this article is to establish a non-integrated defect relation with truncated level for a meromorphic mapping on M intersecting a family of hyperplanes in ${\mathbb{P}}^n(\mathbb{C})$ which is non-subdegenerate with respect to the mapping.     

The Z2-orbifold of the universal affine vertex algebra

Let $\mathfrak{g}$ be a simple, finite-dimensional complex Lie algebra, and let $V^k(\mathfrak{g})$ denote the universal affine vertex algebra associated to $\mathfrak{g}$ at level k. The Cartan involution on $\mathfrak{g}$ lifts to an involution on $V^k(\mathfrak{g})$, and we denote by $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ the orbifold, or fixed-point subalgebra, under this involution. Our main result is an explicit minimal strong finite generating set for $V^k{(\mathfrak{g})}^{{\mathbb{Z}}_2}$ for generic values of k.     

On the equational graphs over finite fields

In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y)=0$ with variables X and Y over a finite field ${\mathbb{F}}_q$ of odd characteristic, we define a digraph by choosing the elements in ${\mathbb{F}}_q$ as vertices and drawing an edge from x to y if and only if $E(x,y)=0$. We call this graph as equational graph. In this paper, we study the equational graph when choosing $E(X,Y)=(Y^2-f(X))(\lambda Y^2-f(X))$ with $f(X)$ a polynomial over ${\mathbb{F}}_q$ and λ a non-square element in ${\mathbb{F}}_q$. We show that if f is a permutation polynomial over ${\mathbb{F}}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials f of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.     

Vector-valued nonstationary Gabor frames

As an extension of Gabor frames, nonstationary Gabor (NSG) frames were recently introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper we generalize the notion of NSG frames from $L^2(\mathbb{R})$ to the vector-valued Hilbert space $L^2(\mathbb{R},{\mathbb{C}}^L)$, and investigate the resulting vector-valued NSG frames. We derive a Walnut's representation of the mixed frame operator, and provide some necessary/sufficient conditions for a vector-valued NSG system to be a frame for $L^2(\mathbb{R},{\mathbb{C}}^L)$. Furthermore, we show the existence of painless vector-valued NSG frames, and of vector-valued NSG frames with fast decaying window functions.