667 research outputs found
On the Skolem Problem and Prime Powers
The Skolem Problem asks, given a linear recurrence sequence , whether
there exists such that . In this paper we consider the
following specialisation of the problem: given in addition ,
determine whether there exists of the form , with
and any prime number, such that .Comment: 13 pages, ISSAC 202
Near-Optimal Complexity Bounds for Fragments of the Skolem Problem
Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial).
In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds
An Application of the Feferman-Vaught Theorem to Automata and Logics for<br> Words over an Infinite Alphabet
We show that a special case of the Feferman-Vaught composition theorem gives
rise to a natural notion of automata for finite words over an infinite
alphabet, with good closure and decidability properties, as well as several
logical characterizations. We also consider a slight extension of the
Feferman-Vaught formalism which allows to express more relations between
component values (such as equality), and prove related decidability results.
From this result we get new classes of decidable logics for words over an
infinite alphabet.Comment: 24 page
Distribution of integral values for the ratio of two linear recurrences
Let and be linear recurrences over a number field , and
let be a finitely generated subring of .
Furthermore, let be the set of positive integers such that
and . Under mild hypothesis,
Corvaja and Zannier proved that has zero asymptotic density. We
prove that
for all , where is a positive integer that can be computed in
terms of and . Assuming the Hardy-Littlewood -tuple conjecture, our
result is optimal except for the term
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