Pell equations for polynomials

Twists and generalized Zolotarev polynomials

“…Indeed it is easy to see that (X, Y ) = (t g+1 (x), 1) satisfies this equation. (Note also that, by [3], this solution is primitive, namely, all the solutions are obtained from this by the multiplication rule in [3].) Moreover, as it is well-known, all the zeros of t n (x) are real and belong to the closed interval [−2, 2].…”

“…In this section we recall the main result of [3] which describes the set of polynomial solutions of a certain type of Diophantine equation in terms of the theory of twists.…”

“…As for the set of all the polynomial solutions of (2.1), the following theorem is proved in [3]: Theorem 2.1 ([3, Theorem 3.1]). Let ∞ 1 , ∞ 2 denote the points above the infinity of C, and put m = ∞ 1 + ∞ 2 .…”

“…Then the group Hom k (J m , G m ) is generated by the element which corresponds to h under the bijection above. Hence the set of all the polynomial solutions of T is generated by h under the multiplication rule described in [3].…”

“…Furthermore, it is pointed out in [10] that, when the base field is the field of complex numbers, ( * ) is related with certain planar graphs through the notion of "dessins d'enfants". In the present paper, generalizing the method in [3], we will see the theory of twists provides us with natural and unified viewpoint to investigate these types of problems for polynomials f (x) of any degree. The plan of this paper is as follows.…”