On the finiteness of hyperelliptic fields with special properties and periodic expansion of \(\sqrt{f}\) (Q1732079)

The objective of this paper is to prove that the set of square-free polynomials $f\in k[x]$ of odd degree distinct from $11$, up to equivalence for which the continued fraction expansion of $\sqrt{f}$ in the complete field $k((x))$ is periodic and the hyperelliptic field $k(x)(\sqrt{f})$ contains an $S$-unit of degree $11$, is finite. For $k=\mathbb{Q}$ do not exist polynomials of odd degree different from $9$ and $11$ satisfying the above conditions. The authors find a new effective method for solving the norm equation and as a consequence it is proved the objective of the paper. The main result establishes that there exists a universal constant $C$ independent of the ground field $k$ that bounds the number of pairwise inequivalent polynomials of odd degree other than $11$ for which the elements $\sqrt{f}$ is periodic in $k((x))$ and the hyperelliptic field $L=k(x)(\sqrt{f})$ contains a fundamental $S$-unit of degree $11$ where $S$ is the set of primes in $L$ above $x$ and it is assumed that $x$ splits in $L$. Furthermore, in the case $\deg f=11$ the explicit family satisfying the above conditions is given. Finally there are no polynomials of $\deg f=1$ or of odd degree $\deg f>11$ satisfying the above conditions. The proof runs as follows. The periodicity of $\sqrt{f}$ implies that the norm equation has a solution. Using some Gröbner basis it is proved that the set of solutions is finite. The problem is reduced to the computation of the Gröbner basis of an equivalent system that is more suitable for computer analysis.