Diophantine approximation with algebraic points of bounded degree. (Q1970620)

The authors extend Schmidt's subspace theorem to the approximation of algebraic numbers by algebraic points of bounded degree. Earlier E. Wirsing proved an extension of Roth's theorem that was generalized by W. M. Schmidt to the approximation of algebraic hyperplanes in projective space by \(k\)-points, \(k\) is a fixed number field. Now here the authors prove as their main theorem a Wirsing-style generalization of Schmidt's theorem. Their method of proof is similar to the so-called ``Stoll's algebroid reduction method'' [cf. \textit{W. Stoll}, Lect. Notes Math. 1277, 131--241 (1987; Zbl 0627.32002)] in Nevanlinna theory and exposed in another paper of \textit{Min Ru} [Math. Z. 233, 137--148 (2000; Zbl 0951.32015)]. Finally they apply their main theorem to solutions of an equation of the form \(F(\mathbf x) = G(\mathbf x),\) where \(F\) and \(G\) are homogeneous forms with \(S\)-integral coefficients in \(n+1\) variables, where \(S\) is a finite set of places containing all the Archimedean places in a number field \(k\). If \(F\) decomposes into a product of linear forms satisfying the strong general position hypothesis of their main theorem, and if \(\deg G < \deg F - 2 {{n+r}\choose r} + 2,\) then there are only finitely many algebraic \(S\)-integer solutions \(\mathbf x\) to \(F(\mathbf x) = G( \mathbf x)\) with \([k(\mathbf x): k]\leq r\).