New Results on the Periodicity Problem for Continued Fractions of Elements of Hyperelliptic Fields

We study the problem of describing square-free polynomials of odd degree with periodic expansion of into a functional continued fraction in , where . We obtain a complete description of such polynomials that does not depend on the field and the degree of a polynomial, provided that the degree of the fundamental -unit of the corresponding hyperelliptic field either does not exceed or is even and does not exceed .