Diophantine approximation in positive characteristic
Let K be a field of characteristic \(p>0\). A classical order is defined on Kx as follows: let \(f(x)=\sum^{\infty}_{n=t}a_ nx^ n\) with \(a_ t\neq 0\). Then one puts \(ord(f)=t\). Mahler then defind \(\alpha (y)=\limsup_{H(r)\to \infty,\quad r\in K(x)}(ord(y-t)/H(r))\) with \(H(P/Q)=\max (\deg P, \deg Q)\). Also for y algebraic over K(x), [K(x,y): K(X)] is denoted by d(y). Mahler proved that \(\alpha\) (y)\(\leq d(y).\) Here the author considers \(y\in Kx\) in the form \((ay^ q+b)/(cy^ q+d)\) with a, b, c, \(d\in Kx\), ad-bc\(\neq 0\) and with q a power of p. The author wants to study the approximation of y by rational functions (with respect to the valuation ord). The main theorem shows that, provided \(d(y)>1\), for any \(r\in K(x),\) either \(H(r)\leq A(q-1),\) or ord(y-1)\(\leq \alpha (y) H(r)+\frac{\alpha (y)A}{q-1}+\frac{B+2 ord(y)}{q-1}\) with \(A=\max (\deg a\), deg b, deg c, deg d) and \(B=ord(ad-bc)\). Applications and examples are given. They show the main theorem is the best possible.
scientific article; zbMATH DE number 4079546
scientific article