Normal continued fractions in a field of hyperelliptic functions (Q2773316)

Suppose \(Y^2=D(X)\), with \(\deg D=2g+2\). Let \(Y=[a_0,a_1,a_2,a_3,\dots]\) denote the continued fraction expansion of \(Y=\sqrt{D(X)}\). Then, other than for \(a_0\), generically the partial quotients \(o_h\) all have degree \(1\), that is, such expansions are `normal'. In any case, a partial quotient has degree at most \(g+1\) and, indeed, the occurrence of \(a_r\) of degree \(g+1\) entails that \(Y\) has a quasi-period of length \(r\). NEWLINENEWLINENEWLINEThis paper discusses several interesting issues arising from unexpected behaviour of the continued fraction expansion of \(f^nY\) for \(n\geq 2\) with \(f\) a polynomial prime to \(D\) which entails in particular that the expansion is not quasi-periodic if the base field is of characteristic zero; and providing a lower bound for the quasi-period in large positive characteristic. By the way, their result may and should be reinterpreted as terms of certain polynomial binary recurrence sequences not having zeros of multiplicity greater than one. NEWLINENEWLINENEWLINEThe authors note that, given \(D\) monic with rational integer coefficients, it makes sense to consider \(D_p=D(X) \pmod p\) as the prime \(p\) varies and hence to ask about the minimal quasi-period \(r_p\) of the continued fraction expansion of \(Y_p\). These issues are discussed in the reviewer's note `Non-periodic continued fractions in hyperelliptic function fields' (Dedicated to George Szekeres on his 90th birthday), Bull. Aust. Math. Soc. 64, 331-343 (2001; Zbl 1013.11036)], and it is, I think, straightforward to confirm their conjecture on the matter from its remarks.