Let (Gn(x)) ∞ n=0 be a d-th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let m ≥ 2 be a given integer. We ask for n ∈ N such that the equation Gn(x) = g • h is satisfied for a polynomial g ∈ C[x] with deg g = m and some polynomial h ∈ C[x] with deg h > 1. We prove that for all but finitely many n these decompositions can be described in "finite terms" coming from a generic decomposition parameterized by an algebraic variety. All data in this description will be shown to be effectively computable.

This chapter details Umberto Zannier's minicourse on hyperelliptic continued fractions and generalized Jacobians. It begins by presenting the Pell equation, which was studied by Indian, and later by Arabic and Greek, mathematicians. The chapter then addresses two questions about continued fractions of algebraic functions. The first concerns the behavior of the solvability of the polynomial Pell equation for families of polynomials. It must be noted that these questions are related to problems of unlikely intersections in families of Jacobians of hyperelliptic curves (or generalized Jacobians). The chapter also reviews several classical definitions and results related to the continued fraction expansion of real numbers and illustrates them by examples.

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In this sequence, solutions (x, y) of the Pell equation x 2 − 3y 2 = 1 occur as numerators and denominators, as one stops the continued fraction at even stages: