Florian Luca

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Florian Luca,
Wits University, Afrique du Sud, et Max Planck Institute for Software Systems, Saarbrücken, Allemagne

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Universal Skolem Sets

The celebrated Skolem-Mahler-Lech theorem asserts that if *u* := (*u*_{n})_{n ≥ 0}
is a linearly recurrent sequence of integers,

then the set of its zeros, that is the set of positive integers *n* such *u*_{n} = 0,
form a union of finitely many infinite arithmetic

progressions together with a (possibly empty) finite set. Except for some special cases, is not known how to bound

effectively all the zeros of *u*. This is called the Skolem problem.
In this talk we present the notion of a universal Skolem

Skolem set, which is an infinite set of positive integers *S* such that for every linearly
recurrent sequence *u*, the solutions

*u*_{n} = 0 with *n* ∈ *S* are effectively computable. We present
a couple of examples of universal Skolem sets, one of which

has positive lower density as a subset of all the positive integers.

It is joint a work with J. Ouaknine (Max Planck Saarbrücken), J. B. Worrell (Oxford).