Related papers: On the characterization of Hilbertian fields

We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many…

Let K be an algebraically closed field. We prove that a polynomial K-derivation $D$ in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is…

Given a finite group $G$ and a number field $K$, we investigate the following question: Does there exist a Galois extension $E/K(t)$ with group $G$ whose set of specializations yields solutions to all Grunwald problems for the group $G$,…

We consider the problem of solvability of linear differential equations over a differential field~$K$. We introduce a class of special differential field extensions, which widely generalizes the classical class of extensions of differential…

We study the irreducible quotient $\mathcal{L}_{t,c}$ of the polynomial representation of the rational Cherednik algebra $\mathcal{H}_{t,c}(S_n,\mathfrak{h})$ of type $A_{n-1}$ over an algebraically closed field of positive characteristic…

We generalize Carlitz' result on the number of self reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula hold for the number of irreducible polynomials obtained by a fixed quadratic…

Let $D$ be a negative integer congruent to $0$ or $1\bmod{4}$ and $\mathcal{O}=\mathcal{O}_D$ be the corresponding order of $ K=\mathbb{Q}(\sqrt{D})$. The Hilbert class polynomial $H_D(x)$ is the minimal polynomial of the $j$-invariant $…

We investigate the standard graded $k$-algebras over a field $k$ of characteristic zero for which general linear forms are exact zero divisors. We formulate a conjecture regarding the Hilbert function of such rings. We prove our conjecture…

We find geometric and arithmetic conditions in order to characterize the irreducibility of the determinant of the generic Vandermonde matrix over the algebraic closure of any field k. We also characterize those determinants whose…

The aim of this paper is to present an explicit reduction algorithm for Hilbert modular groups over arbitrary totally real number fields. An implementation of the algorithm is available to download from [19]. The exposition is…

This paper aims to describe the restricted Kac modules of restricted Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p>3$. In particular, a sufficient and necessary condition for the…

We introduce symmetrizing operators of the polynomial ring $A[x]$ in the varible $x$ over a ring $A$. When $A$ is an algebra over a field $k$ these operators are used to characterize the monic polynomials $F(x)$ of degree $n$ in $A[x]$ such…

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\…

We show that an infinite group $G$ definable in a $1$-h-minimal field admits a strictly $K$-differentiable structure with respect to which $G$ is a (weak) Lie group, and show that definable local subgroups sharing the same Lie algebra have…

We consider a 2-dimensional representation of the Hecke algebra $\mathcal{H}(G_7, u)$, where $G_7$ is the complex reflection group and $u$ is the set of indeterminates $u=(x_1, x_2, y_1, y_2, y_3, z_1, z_2, z_3)$. After specializing the…

We discuss, using the Hilbert basis method, how to efficiently construct a complete basis for D-flat directions in supersymmetric Abelian and non-Abelian gauge theories. We extend the method to discrete (R and non-R) symmetries. This…

We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of…

We consider the problem of defining polynomials over function fields of positive characteristic. Among other results, we show that the following assertions are true. 1. Let $\G_p$ be an algebraic extension of a field of $p$ elements and…

Let R be an affine k-domain over the field k. The paper's main result is that, if R admits a non-trivial embedding in a polynomial ring K[s] for some field K containing k, then R can be embedded in a polynomial ring F[t] which extends R…

A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for certain families of arrangements which…