On the parametrization of hyperelliptic fields with \(S\)-units of degrees 7 and 9 (Q2090545)

Based on authors' abstract: This paper shows that if \(k\) is an algebraically closed field with \(\operatorname{char}k=0\), then the set of polynomials \(f\) of degree 5 such that the field \(k(x)(\sqrt{f}\,)\) has a nontrivial \(S\)-unit of degree 7 or 9 and the continued fraction expansion of \(\sqrt{f}/x\) is periodic is a one-parameter set corresponding to a rational curve with finitely many deleted points. In this paper the authors established two theorems and explain its proofs with the help of different types of proposition and definition.