Jakub Konieczny

Finitely-valued generalised polynomials

Generalised polynomials are expressions constructed from polynomials with the use of the floor function, addition and multiplication, such as [√2n [√3n2] + √6n + 1/2]. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for ordinary polynomials. In particular, there exist generalised polynomial sequences which take only finitely many values but are not constant. This is the case, for instance, for Sturmian sequences. In my talk, I will discuss such finitely-valued generalised polynomial sequences, their connection with nilpotent dynamics, estimates on their subword complexity, constructions of potentially surprising examples, as well as problems concerning decidability. The talk will include a survey of existing results on generalised polynomials, adapted to the current context, as well as several new results obtained in joint works with Boris Adamczewski and with Jakub Byszewski.