On the finiteness of hyperelliptic fields with special properties and periodic expansion of \(\sqrt{f}\)
type: Item
instance of: scholarly article
title: On the finiteness of hyperelliptic fields with special properties and periodic expansion of \(\sqrt{f}\)
zbMATH Open document ID: 1428.11131
DOI: 10.1134/S1064562418070281
author: Vladimir S. Zhgoon
author: M. M. Petrunin
author: V. P. Platonov
author: Yu. N. Shteĭnikov
published in: Doklady Mathematics
publication date: 2019-03-15T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 7036544
zbMATH Keywords: hyperelliptic fields
zbMATH Keywords: continued fractions
zbMATH Keywords: $S$-unit
zbMATH Keywords: periodic elements
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
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full work available at URL: s1064562418070281
OpenAlex ID: W2910239657
description: scientific article; zbMATH DE number 7036544