Seminarium Nieliniowość i Geometria

In recent years, stating that a rational map is `integrable', has become synonymous with saying that its iterates exhibit `polynomial degree growth' in terms of the initial conditions. For higher-dimensional maps, however, it is in general very difficult to calculate the degrees of their iterates in any systematic way and there is a pressing need to develope reliable, rigorous, algorithms to this end.In this seminar, I present two such calculation methods that can be applied to rational maps of any order. Both methods are based on the singularity structure of the map, one is a method that rigorously calculates the algebraic entropy of the map while the other provides an ultra-efficient algorithm for generating the degree sequence for the iterates of the map.
This talk is (partly) based on on [A. Stokes, T. Mase, R. Willox and B. Grammaticos, "Deautonomisation by singularity confinement and degree growth", The Journal of Geometric Analysis 35, 65 (2025)] and [B. Grammaticos, A. Ramani, A.S. Carstea and R. Willox, "A Fast Algorithmic Way to Calculate the Degree Growth of Birational Mappings", Mathematics 13, no. 5: 737 (2025)]