Enumerative problem for Pell–Abel equation

The Pell–Abel (PA) functional equation is the reincarnation of the famous Diophantine equation in the world of polynomials, considered N. H. Abel in 1826. The equation arises in many problems: reduction of Abelian integrals, elliptic billiards, the spectral problem for infinite Jacobi matrices, approximation theory, etc. If the PA equation has a nontrivial solution, then there are infinitely many of them, and all of them are expressed via a primitive solution having a minimum degree. Using graphical techniques, we find the number of connected components in the space of PA equations with the coefficient of a given degree and having a primitive solution of another given degree.
Joint work  with Quentin Gendron (UNAM Institute of Mathematics) https://arxiv.org/abs/2306.00884.