Related papers: On the existence of dimension zero divisors in alg…

The main purpose of this paper is to prove the following theorem on the defect relations for ample divisors of abelian varieties. Main Theorem. Let $A$ be an abelian variety of complex dimension $n$ and $D$ be an ample divisor in $A$. Let…

There exists a function f: N -> N such that for every positive integer d, every quasi-finite field K and every projective hypersurface X of degree d and dimension at least f(d), the set X(K) is non-empty. This is a special case of a more…

We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small $g$-heights with respect to…

Some properties of $m$-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \begin{itemize} \item {\it Let $\lambda$ be a continuous differential form of degree $h$ in…

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $z\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. Set $\alpha _0~:=~\sqrt{3}/2\pi ~=~0.2756644477\ldots$. We…

The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of the…

In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums. Precisely, let $h=\lfloor q^{\delta}\rfloor>1$ and $d\mid q^h-1$. Let $r$ be…

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field $L$ has infinitely many nontrivial zeros of multiplicity at least 2 if $L$ has a subfield $K$ for which $L/K$ is a…

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…

We fix a field $\kk$ of characteristic $p$. For a finite group $G$ denote by $\delta(G)$ and $\sigma(G)$ respectively the minimal number $d$, such that for any finite dimensional representation $V$ of $G$ over $\kk$ and any $v\in…

In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset $\mathcal{E}$ in a regular variety satisfies $|\mathcal{E}|\gg…

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that…

We develop the theory of difference algebraic groups in the case where we have finitely many pairwise commuting difference operators. We show that the defining ideal of a difference algebraic group is finitely generated as a difference…

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j…

Let $G$ be a linear algebraic group over a field $k$, and let $V$ be a $G$-module. Recall that the nullcone of $(G,V)$ is the set of points $v$ in $V$ with the property that $f(v)=0$ for every positive degree homogeneous invariant $f$ in…

A new simple way to prove the Frobenius conjecture on the dimensions of real algebras without zero divisors is given.

Let $A$ be an integral domain with quotient field $K$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra. Denote by $D(F)$ the discriminant of a polynomial $F\in A[X]$. Further, given a finite etale algebra $\Omega$,…

A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of…

Let $\mathbb{F}_q$ be the finite field of $q$ elements. In this paper we obtain bounds on the following counting problem: given a polynomial $f(x)\in \mathbb{F}_q[x]$ of degree $k+m$ and a non-negative integer $r$, count the number of…