Twisted Rational Zeros and Local-Global Principles for Linear Recurrence Sequences

In 1934, Skolem conjectured the exponential local-global principle; that a linear recurrence sequence (LRS) has an integer zero if and only if it has a zero modulo every integer m. 

An interesting counterexample shows that a natural strengthening of the Skolem Conjecture, where we restrict m to be a prime power, is false. The failure in this example is accounted for by the relationship between p-adic zeros of LRS and twisted rational zeros (introduced by Bilu et al. (2025)).

I will talk about some results which may be extracted by studying this relationship further, in particular, a local-global principle for simultaneous zeros of two coprime LRS, subject to the p-adic Schanuel Conjecture. A fundamental step in proving this result is characterising when twisted rational zeros are (or are not) p-adic zeros for infinitely many primes p, which may be done unconditionally. This addresses two questions of Bilu et al. (2025).