Hyperelliptic Curves, Continued Fractions and Somos Sequences

In the course of studying continued fraction expansions
(Z + Ph)/Qh = ah − (Z + Ph+1)/Qh , h ∈ Z
in quadratic function fields I learned that the leading coefficients, say
dh , of the polynomials Ph control the degeneracies of the expansion.
So I chose to try to describe the other parameters detailing the Ph and
Qh in terms of the dh .
Denote a typical zero of Qh by ϑh and recall the recursion relations
Ph + Ph+1 + A = ahQh and
− QhQh+1 = (Z + Ph+1)(Z + Ph+1) = −R + Ph+1(A + Ph+1) .
Thus Ph(ϑh) + Ph+1(ϑh) + A(ϑh) = 0, R(ϑh) = −Ph+1(ϑh)Ph(ϑh).
Hence Qh(X) divides R(X) + Ph+1(X)Ph(X), and so
Ch(X)/uh = ` R(X) + Ph+1(X)Ph(X) ´ /Qh(X)
defines a polynomial Ch . Here uh is the leading coefficient of Qh .
It’s useful that deg Ch = max ` g, 2(g − 1) ´ − g ; so Ch is a constant if
g = 1 or g = 2.
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