Diophantine approximation in characteristic \(p\)
Let \(K_v\) be the completion of a global field \(K\) of positive characteristic \(p\), with respect to a valuation \(v\) of \(K\). For \(y\in K_v \setminus K\), the author defines \(\alpha (y)= \limsup_{r\in K} v(y- r)/ h(r)\), where \(h(r)= [K: k(r)]\) and \(k\) is the constant field of \(K\). Also he introduces for every real \(\alpha\), \(b(y, \alpha)= \limsup_{r\in K} (v(y- r)- \alpha h(r))\). The first result is the following: let \(y\) be an element of \(K_v\) satisfying \(y^q= (ay+ b)/ (cy+ d)\), where \(a,b,c,d\in K\), \(ad- bc\neq 0\), and \(q\) is a power of \(p\). Let \(R(Y)\) be a rational fraction of degree \(d\) in \(Y\). If \(\alpha (y)> d(q^{1/2}+ 1)\), then \(\alpha (R (y))= \alpha (y)/ d\) and \(b(R(y), \alpha (R(y)) \neq\pm \infty\) (of course, in this statement, the letter \(d\) has two distinct meanings!). Theorem 2 gives an interesting example of \(s\) algebraic over \(K\) with \(b(s, \alpha (s))= +\infty\), where \(K\) is the function field of an elliptic curve defined over a finite field \(k\). The author gives a method to obtain such examples with \(K= k(x)\), but gives effectively no example. Lastly, Theorem 3 asserts that the cross ratio of any four conjugates of an algebraic element \(y\in K_v\) such that \(b(y, [(d+ 3)/2])= +\infty\) lies in \(k\). This result is not absolutely exact (take \((1+ 1/x)^{1/d}\) in \(\mathbb{F}_p (x)\) with \((d,p) =1\)). This cross ratio lies actually in the constant field of \(K(y, y', y'', y''')\); indeed, there is a slight error on page 325, line 5, where \(K\) should be replaced by \(K(y, y', y'', y''')\). On the other hand, Osgood's method actually permits to put \([d/2+ 1]\) in place of \([(d+3)/ 2]\) [see also \textit{W. M. Schmidt}, Commun. Pure Appl. Math. 29, 759-773 (1976; Zbl 0333.10019)].
scientific article; zbMATH DE number 757761