Séminaire TNGA : Gessica Alecci

Oratrice : Gessica Alecci (Turin)

Titre : On the algebraic structure of linear recurrence sequences and on Lucas atoms

Résumé : In the first part of my talk, I will show some results about the algebraic structures of the set of linear recurrences with several operations, such as the binomial convolution or the multinomial convolution. In particular, we will see in which cases we have R-Algebras and explore whether these R Algebras are isomorphic. In the second part, I will talk about a joined work with P. Miska, N. Murru, and G. Romeo about the Lucas atoms. In 2020, Sagan and Tirrell introduced these objects, which are irreducible factors of Lucas polynomials. We defined them in a more natural and powerful way than the original definition, providing straightforward proofs of their main properties. Moreover, we fully characterize the p-adic valuations of Lucas atoms for any prime p, answering to a problem left open by Sagan and Tirrell. Finally, we prove that the sequence of Lucas atoms is not holonomic, in contrast to the Lucas sequence, that is a linear recurrent sequence of order two.