Links between Mason's theorem, Belyi descent and different versions of the \((abc)\)-conjecture
The author describes some developments in which he has been involved in connection with the \(ABC\)-theorem of R. C. Mason and the associated \(abc\)-conjecture. The \(abc\)-conjecture is extended to apparently more general statements concerning the radical (product of the distinct prime factors) of \(abF(a,b)\) where \(F(a,b)\) need not be \(a+b\), or even to one concerning a homogeneous polynomial \(G(a,b)\) not necessarily of the form \(abF(a,b)\). However, it is a consequence of a certain local form of Mason's theorem that these general conjectures are in fact implied by the usual \(abc\)-conjecture. Some consequences of the conjecture, for example concerning the radical of \(a^3-b^2\), are also discussed. Attention is given to examples where Mason's bound for the number of complex roots of a polynomial \(AB(A+B)\) is attained. These are, for example, of importance in connection with results of \textit{F. W. Beukers }[Duke Math. J. 91, No. 1, 61-88 (1998; Zbl 1038.11505)] concerning \(x^5+y^3=z^2\) and other Diophantine equations. A forthcoming paper of the author will contain a related analysis of the equations \(b^2-c^3=\pm a^4\). An example found by A. Nitaj provided an identity of degree 48 in which the irreducible factors of \(AB(A+B)\) are of degree no larger than 4. This permitted an improvement [\textit{G. Greaves}, and \textit{A. Nitaj}, Number theory in progress, Vol. 1, 229-236 (W. de Gruyter, Berlin, 1999; Zbl 0933.11016)] in a bound related to the \(abc\)-conjecture obtained in [\textit{J. Browkin, M. Filaseta, G. Greaves}, and \textit{A. Schinzel}, Lond. Math. Soc. Lect. Note Ser. 237, 65-85 (1997; Zbl 0926.11022)].
scientific article; zbMATH DE number 1370106