The number of components of the Pell-Abel equations with primitive solutions of given degree. (English. Russian original) Zbl 1523.30012

Bogatyrev, Andrei B.; Gendron, Quentin

Russ. Math. Surv. 78, No. 1, 208-210 (2023); translation from Usp. Mat. Nauk 78, No. 1, 209-210 (2023).

From the introduction: N. H. Abel [J. Reine Angew. Math. 1, 185–221 (1826; ERAM 001.0021cj)] considered Pell’s Diophantine equation over the ring of polynomials. Since then the equation \[ P^2(x)-D(x)Q^2(x)=1 \]bears the names of both Pell and Abel. Here \(P (x)\) and \(Q(x)\) are unknown polynomials of one variable and \(D(x) := \prod_{e\in E}(x-e)\) is a given monic complex polynomial of degree \(\mathrm{deg} D = |E| := 2g + 2\) without multiple roots. [\(\ldots\)]
The main result of this note consists in finding the number of connected components of the complex Pell-Abel equations with polynomial \(D\) of fixed degree \(2g + 2\) which admit a primitive solution \(P\) of another fixed degree \(n\). [\(\ldots\)]

MSC:

30C10	Polynomials and rational functions of one complex variable
34A05	Explicit solutions, first integrals of ordinary differential equations

Keywords:

complex polynomials; Pell-Abel equation

Citations:

ERAM 001.0021cj