The complement of Fermat curves in the plane (Q399453)

The paper under review proves some upper bounds for the height of integral points over function fields. Specifically, the integral points are on the projective plane minus the union of the line \(z=0\) and the Fermat curve \(x^n+y^n=z^n\) of degree \(n\). The main theorem bounds from above the height of an integral point outside an explicit exceptional set by the following quantity: NEWLINE\[NEWLINE\frac{3(m+\max\{2g-2,0\})}{n-4}NEWLINE\]NEWLINE where \(m\) is the number of places of the function field at which the coordinates of the integral point have poles, \(g\) is the genus of the function field, and \(n\) is the degree of the Fermat curve being deleted.NEWLINENEWLINEThe main tool used in the proof is a theorem of \textit{W. D. Brownawell} and \textit{D. W. Masser} [Math. Proc. Camb. Philos. Soc. 100, 427--434 (1986; Zbl 0612.10010), Theorem B], which is a version of the \(abc\) conjecture for function fields.