Positivity certificates for linear recurrences
arXiv:2306.05930 · doi:10.1137/1.9781611977912.37
Abstract
We consider linear recurrences with polynomial coefficients of Poincaré type and with a unique simple dominant eigenvalue. We give an algorithm that proves or disproves positivity of solutions provided the initial conditions satisfy a precisely defined genericity condition. For positive sequences, the algorithm produces a certificate of positivity that is a data-structure for a proof by induction. This induction works by showing that an explicitly computed cone is contracted by the iteration of the recurrence.
18 pages. To appear in Proceedings SODA'24