Title: Positivity of P-Recursive Sequences Satisfying Linear Recurrences
| Speaker | Steve Melczer |
| Affiliation | University of Waterloo |
| Location | MC 5479 |
Abstract: Whether it is decidable to determine when sequences satisfying linear recurrences with constant coefficients have all positive terms is a notorious problem in enumerative combinatorics that has essentially been open for around 90 years. Nevertheless, a "meta-principle" states that all such sequences arising from combinatorial counting problems belong to a special class where positivity (and more general asymptotic
behaviour) is decidable. Here we discuss new software for determining positivity for sequences satisfying linear recurrences with *polynomial* coefficients. Originally motivated by a novel approach to proving genus one solution uniqueness for the Canham model for biomembrane shapes, our algorithm combines rigorous numeric analytic continuation of functions satisfying linear ODEs with singularity analysis techniques from analytic combinatorics. The main talk will be presented using a live Sage Jupyter notebook, and audience members who have access to Sage with a recent version of the ore_algebra package installed (available at
https://github.com/mkauers/ore_algebra) will be able to follow along and play with the package during the talk.
There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,