Related papers: A generalised Skolem-Mahler-Lech theorem for affin…

Let X be a separated scheme of finite type over an algebraically closed field k and let m be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod m Suslin homology and the first…

This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G with a subdegree-finite, infinite…

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic…

In this paper, we introduce a condition $\mathrm{(F}_m'\mathrm{)}$ on a field $K$, for a positive integer $m$, that generalizes Serre's condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules…

Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime.…

An algebra $S$ is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra $R$, every homomorphism $R\to R\otimes S$ extends to an inner automorphism of $R\otimes S$. One of the important properties of such…

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…

In this paper we generalise the notion of linearity (in the sense of Lawvere) to a category C equipped with a compatible sum structure and product structure. In this context, any morphism f from an n-fold sum to an n-fold product has a…

We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also…

We obtain several finiteness results for the unramified cohomology of function fields of algebraic varieties defined over fields of type (F'_m), a class that includes algebraically closed fields, finite fields, local fields, and some higher…

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then…

In this paper we prove a new generic vanishing theorem for $X$ a complete homogeneous variety with respect to an action of a connected algebraic group. Let $A, B_0\subset X$ be locally closed affine subvarieties, and assume that $B_0$ is…

We prove a monodromy theorem for local vector fields belonging to a sheaf satisfying the unique continuation property. In particular, in the case of admissible regular sheaves of local fields defined on a simply connected manifold, we…

A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for infinite index "congruence" subgroups of SL(2,Z).…

Let $F(s)=\sum_n a_n/\lambda_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely…

A conjecture of J. Huh and B. Sturmfels predicts that the sign of the Euler characteristic of a complex very affine variety depends only on the parity of the dimension. The conjecture is true for locally complete intersections. Beyond this…

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha…

A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts $S$ and a fixed…

It is well known that ZFC, despite its usefulness as a foundational theory for mathematics, has two unwanted features: it cannot be written down explicitly due to its infinitely many axioms, and it has a countable model due to the…

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F)…