Differential equations and Diophantine approximation in positive characteristic
The subject is diophantine approximation of formal Laurent series \(F=\sum_{n=-m}^\infty f_nX^{-n}\) over some field \(K\); here \(\deg F=\deg_X F=m\). Let \(P\) and \(Q\neq 0\) denote polynomials in \(X\). It is well known that Roth's Theorem fails in positive characteristic and indeed that Liouvile's Theorem is best possible. However the authors remark that they have shown [J. Reine Angew. Math. 473, 195-206 (1996; Zbl 0844.11046)] that Thue's method leads to \(\deg(F-P/Q)> -(\frac 12\lfloor r\rfloor +1 +\varepsilon)\deg Q\), for algebraic elements \(F\) of degree \(r\), provided that \(F\) is not of the shape \(F=(AF^{p^s}+B)/(CF^{p^s}+D)\) for polynomials \(A\), \(B\), \(C\), and \(D\). In the present note the authors first detail a version of \textit{C. F. Osgood}'s argument [Nederl. Akad. Wet, Proc., Ser. A 78, 401 (1975; Zbl 0313.10036)] showing that, if \(F\) does not satisfy a rational Riccati differential equation, the arbitrary \(\varepsilon>0\) may be omitted above (up to an \(O(1)\) correction). They then apply a remark of \textit{J. F. Voloch} [Monatsh. Math. 119, 321-325 (Zbl 0827.11039)] to obtain their main result that if \(K\) is a finite field the \(\varepsilon\) may also be omitted under their condition. In passing one sees that if a rational Riccati equation has no rational solution then a solution \(F\in K((1/X))\) satisfies \(\deg(F-P/Q)> -2\deg Q-O(1)\). The authors note that, in characteristic \(2\), there are rational Riccati equations without a rational solution. This is not so in characteristic \(p=3\). The cases \(p>3\) remain open.
diophantine approximation of formal Laurent series
scientific article; zbMATH DE number 1340540