University of Birmingham > Talks@bham > Theoretical computer science seminar > Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

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If you have a question about this talk, please contact Rajesh Chitnis.

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 (up to 4) or by restricting the structure of the LRS e.g., roots of the characteristic polynomial of the LRS . We identify a large subclass of LRS (LRS whose characteristic roots are possibly complex roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic) for which the Skolem problem is relatively easy—it is (almost) NP-complete. 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.

This talk is part of the Theoretical computer science seminar series.

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