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

This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a…

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem-Mahler-Lech…

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem,…

The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory.…

We show that in a parametric family of linear recurrence sequences $a_1(\alpha) f_1(\alpha)^n + \ldots + a_k(\alpha) f_k(\alpha)^n$ with the coefficients $a_i$ and characteristic roots $f_i$, $i=1, \ldots,k$, given by rational functions…

Let X be a Noetherian space, let f be a continuous self-map on X, let Y be a closed subset of X, and let x be a point on X. We show that the set S consisting of all nonnegative integers n such that f^n(x) is in Y is a union of at most…

The Skolem Problem asks to determine whether a given integer linear recurrence sequence has a zero term. This problem arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory,…

We propose a generalization of Schramm-Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie…

We prove that if $\mathbb{F}$ is an algebraically closed field of zero characteristic which has infinite transcendence degree over $\mathbb{Q}$, then there exists a field automorphism $\varphi$ of ${\rm SL}_n(\mathbb{F})$ and ${\rm…

We show that the Skolem Problem is decidable in finitely generated commutative rings of positive characteristic. More precisely, we show that there exists an algorithm which, given a finite presentation of a (unitary) commutative ring…

We present a relatively simple inductive proof of the classical Levy-Steinitz Theorem saying that for a sequence $(x_n)_{n=1}^\infty$ in a finite-dimensional Banach space $X$ the set of all sums of rearranged series $\sum_{n=1}^\infty…

Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the L\"owenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit…

An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup…

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a…

A generalized polymorphism of a predicate $P \subseteq \{0,1\}^m$ is a tuple of functions $f_1,\dots,f_m\colon \{0,1\}^n \to \{0,1\}$ satisfying the following property: If $x^{(1)},\dots,x^{(m)} \in \{0,1\}^n$ are such that…

Let $\Gamma =(V,E)$ be a reflexive relation having a transitive group of automorphisms and let $v\in V.$ Let $F$ be a subset of $V$ with $F\cap \Gamma ^-(v)=\{v\}$. (i) If $F$ is finite, then $| \Gamma (F)\setminus F|\ge |\Gamma (v)|-1.$…

Motivated by the famous Skolem-Mahler-Lech theorem we initiate in this paper the study of a natural class of determinantal varieties which we call {\em Vandermonde varieties}. They are closely related to the varieties consisting of all…

Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists a $\gamma>0$ such that any subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial…

Levy-Steinitz theorem characterize sum range of conditionally convergent series, that is a set of all its convergent rearrangements; in finitely dimensional spaces -- it is an affine subspace. An achievement of a series is a set of all its…