Groups of \(S\)-units in hyperelliptic fields (Q957818)

Let \(k={\mathbb F}_ q(x)\) be a congruence rational function field over the finite field of \(q\) elements of characteristic \(p>2\). For a square free polynomial \(d(x)=a_ {2n+1}x ^ {2n+1}+ a_ {2n}^ {2n}x^{2n}+\cdots+a_ 0\) with \(a_{2n+1}\neq 0\), let \(K=k(\sqrt{d})\) be a hyperelliptic function field over \({\mathbb F}_ q\). Let \(S\) be a finite non-empty set of places of \(K\) and let \(U_ S\cong {\mathbb F}_ q ^ * \times {\mathbb Z}^{| S| -1}\) be the group of \(S\) units in \(K\). In the number field case, when \(L={\mathbb Q}(\sqrt{d})\), \(d>0\), is a quadratic extension of the field of rational numbers, a fundamental unit of \(L\) can be found by expanding \(\sqrt{d}\) into a continued fraction. For function fields, the method of continued fractions does not always yield a fundamental unit. The main goal of the paper under review is to construct an algorithm for calculating fundamental \(S\)-units of \(K\). The key fact in finding a system of independent \(S\)-units is to consider the minimal case: \(S=\{w_ \infty, w\}\) where \(w_{| k}=v\) and \(v=ww'\) splits. To find a fundamental \(S\)-unit, it must be found the minimum positive integer \(m\) for which \(f^ 2-g^ 2 d= av^ m\) has a solution for \(a\in{\mathbb F}_ q^ *\) and polynomials \(f,g\in{\mathbb F}_ q[x]\).