Solvable Pell-Abel equations

The Pell–Abel (PA) functional equation $P^2(x)-D(x)Q^2(x)=1$ is the reincarnation of the famous Diophantine equation in the world of polynomials, which was considered by N.H. Abel in 1826. The equation arises in various math environments: 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 $P(x)$ which has a minimal complexity. Using graphical techniques, we find the number of connected components in the space of PA equations with the coefficient $D(x)$ of a given degree and having a primitive solution $P(x)$ of another given degree. Some related problems will be discussed also. Joint work with Quentin Gendron (UNAM Institute of Mathematics)