667 research outputs found

    On the Skolem Problem and Prime Powers

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    The Skolem Problem asks, given a linear recurrence sequence (un)(u_n), whether there exists nNn\in\mathbb{N} such that un=0u_n=0. In this paper we consider the following specialisation of the problem: given in addition cNc\in\mathbb{N}, determine whether there exists nNn\in\mathbb{N} of the form n=lpkn=lp^k, with k,lck,l\leq c and pp any prime number, such that un=0u_n=0.Comment: 13 pages, ISSAC 202

    Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

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    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

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    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

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    Let FF and GG be linear recurrences over a number field K\mathbb{K}, and let R\mathfrak{R} be a finitely generated subring of K\mathbb{K}. Furthermore, let N\mathcal{N} be the set of positive integers nn such that G(n)0G(n) \neq 0 and F(n)/G(n)RF(n) / G(n) \in \mathfrak{R}. Under mild hypothesis, Corvaja and Zannier proved that N\mathcal{N} has zero asymptotic density. We prove that #(N[1,x])x(loglogx/logx)h\#(\mathcal{N} \cap [1, x]) \ll x \cdot (\log\log x / \log x)^h for all x3x \geq 3, where hh is a positive integer that can be computed in terms of FF and GG. Assuming the Hardy-Littlewood kk-tuple conjecture, our result is optimal except for the term loglogx\log \log x

    Universal {S}kolem Sets

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