Related papers: Ray class invariants over imaginary quadratic fiel…

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak…

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…

We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method…

In this paper, we prove the conjecture of Yui and Zagier concerning the factorization of the resultants of minimal polynomials of Weber class invariants. The novelty of our approach is to systematically express differences of certain Weber…

Lehmer constructs four classes of matrices constructed from roots of unity for which the characteristic polynomials and the $k$-th powers can be determined explicitly. Here we study a class of matrices which arise naturally in…

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group…

Let $G$ be a reductive group acting on a path algebra $kQ$ as automorphisms. We assume that $G$ admits a graded polynomial representation theory, and the action is polynomial. We describe the quiver $Q_G$ of the smash product algebra $kQ\#…

The First Fundamental Theorem of Invariant Theory describes a minimal generating set of the invariant polynomial ring under the action of some group $G$. In this note we give an elementary and direct proof for the $\operatorname{GL}_2(K)$…

We introduce a new version $kk^{\rm alg}$ of bivariant $K$-theory that is defined on the category of all locally convex algebras. A motivating example is the Weyl algebra $W$, i.e. the algebra generated by two elements satisfying the…

Let $q$ be a positve integer, and $G$ be a $q$-partite simple graph on $qn$ vertices, with $n$ vertices in each vertex class. Let $\delta={k_q \over k_q+1}$, where $k_q=q+O(\log{q})$. If each vertex of $G$ is adjacent to at least $\delta n$…

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the…

We establish an effective version of Siegel's lower bounds for class numbers of imaginary quadratic fields in certain cures in $Y(1)^n$. Our proof goes through the G-functions method of Yves Andr\'e.

We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

The article presents results on the well-known problem concerning the structure of integer polynomials $p_n(z; x, y)$, which define multiplication laws in $n$-valued groups $\mathbb{G}_n$ over the field of complex numbers $\mathbb{C}$. We…

This is the second in a series of two papers presenting a solution to Hilbert's 12th problem for real quadratic function fields in positive characteristic, in the sense of proving an analog of the Theorem of Weber-Fueter. We also offer a…

For an arbitrary representation $\rho$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $\rho$. Among other interesting properties, these numbers provide lower…

The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…

Let $L/K$ be a finite extension of congruence function fields. We say that $L/K$ is a {\it radical extension} if $L$ is generated by roots of polynomials $u^{M}-\alpha \in K[u]$, where $u^{M}$ is the action of Carlitz-Hayes. We study a…

We show that for any square-free natural number $n$ and any global field $K$ with $(\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\in K^*\times K^*$ such that $x$ is not a norm of $K(\sqrt[n]{y})/K$ form a…