On generalized Jacobians and rational continued fractions in the hyperelliptic fields

In the paper we introduce generalized Mumford polynomials describing additive law on generalized Jacobian of singular hyperelliptic curve over the field $\mathbb K$ of characteristics different from $2$, and smooth at infinity and defined in the affine chart by the equation $y^2=\phi(x)^2f(x)$, where $f$ is a square-free polynomial. We describe the relation between the continued fraction expansion of the quadratic irrationalities in the hyperelliptic function field $\mathbb K(x,\sqrt{f(x)})$ and the generalized Mumford polynomials describing the additive law in the divisor class group of the singular hyperelliptic curve. This correspondence between the continued fraction expansion of the quadratic irrationalities and the generalized Mumford polynomials allow us to prove the theorem on equivalence of two conditions: the condition $(i)$ of quasi-periodicity of continued fraction expansion (related with valuation of a point of degree $1$ on the normalization of the curve) of a quadratic irrationality of the special type and the condition $(ii)$ of the finiteness of the order of the class, related to the point of degree $1$ on the normalization of the curve. By means of this correspondence we also obtain the results on the symmetry of quasi-period and we give estimates for its length, generalizing results obtained before by the author and collaborators.