Pisot and Salem numbers and generalised polynomials

par Jakub Byszewski (Jagiellonian University)

mardi 18 avr. 2023, 10:45 → 11:45 Europe/Paris

Salle Fokko du Cloux, Bât Braconnier (ICJ, Université Lyon 1)

Generalised polynomials are polynomial-like expressions that additionally allow the use of the floor function. They can be studied using ergodic theory---in fact, they arise by evaluating a piecewise polynomial function along an orbit on a nilmanifold.

 

In an earlier work with Jakub Konieczny we noticed that you can find a generalised polynomial expression that vanishes precisely at the Fibonacci numbers. Which other linear recurrence sequences can you realise in that manner? This is the case for linear recurrences with characteristic polynomial the minimal polynomial of a: i) Pisot unit of degree 2; ii) Pisot unit of degree 3 that is not totally real; and iii) Salem number. We expect that the above list is essentially complete (except for some cheap tricks). However, proving that you cannot realise something is considerably more difficult, essentially the only case known being powers of an integer k>=2. We introduce the notion of a generalised polynomial on a number field and we use tools from diophantine geometry to prove a result that will hopefully enable us to extend this further. The talk is based on joint work with Jakub Konieczny.