Description
The Pell–Abel (PA) functional equation P^2-DQ^2=1 is a reincarnation of the famous Diophantine equation in the world of polynomials, considered by N.H. Abel in 1826. The equation arises in many problems: the reduction of Abelian integrals, elliptic billiards, the spectral problem for infinite Jacobi matrices, approximation theory, and so on. If an PA equation has a nontrivial solution (P,Q)\neq (1,0), then there are infinitely many of them, and all of them are expressible in terms of a primitive solution of minimal degree. Using a graphical technique, we find the number of connected components in the space of PA equations with a coefficient D(x) of a given degree and having a primitive solution P(x) of another given degree.
Joint work with Quentin Gendron (Institute of Mathematics, UNAM)
“The space of solvable Pell-Abel equations”, Compositio Mathematica, 161:7 (2025)
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