An \(abcd\) theorem over function fields and applications
In the paper under review, the authors give a lower bound for the number of distinct zeros of the sum \(1+u+v\), where \(u\) and \(v\) are rational functions which is sharp when \(u\) and \(v\) have few distinct zeros and poles compared to their degrees. Their main result sharpens the ``\(abcd\)'' theorem of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results for Diophantine equations with polynomials. The main tool is a prior result of the authors from [``Some cases on Vojta's conjecture on integral points over finite fields'', J. Algebr. Geom. 17, No. 2, 295--333 (2008); addendum Asian J. Math. 14, No. 4, 581--584 (2010; Zbl 1221.11146)]. As applications, they obtain that the Fermat surface \(x^a+y^b+z^c=1\) contains only finitely many rational or elliptic curves when \(a\geq 10^4\) and \(c\geq 2\). They also obtain an interesting application to the so--called ``Diophantine \(k\)-tuples''. Namely, if \(a,~b,~c\) are three distinct nonzero complex polynomials not all constant such that \(1+ab=x^p,~1+ac=y^q\) and \(1+bc=z^r\) with complex polynomials \(x,y,z\) and integers \(p,q,r\geq 864\), then after suitably permuting \(a,b,c\), we have \(c^2+1=0\) and \(a+b=2c\).
scientific article; zbMATH DE number 6024209