5 papers
The exceptional set in the abc conjecture
Christian Bernert
We study the size of the exceptional set in the conjecture, improving on and simplifying work of Browning, Lichtman and Teräväinen.
Integral points of bounded height on quintic del Pezzo surfaces over number fields
Christian Bernert, Ulrich Derenthal
We prove an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to…
Points of bounded height on quintic del Pezzo surfaces over the rational numbers
Christian Bernert, Ulrich Derenthal
We give a relatively short and elementary proof of Manin's conjecture for split smooth quintic del Pezzo surfaces over the rational numbers.
Small solutions to homogeneous and inhomogeneous cubic equations
Christian Bernert
We study the solubility of cubic equations over the integers. Assuming a necessary congruence condition, the existence of such solutions is established when the -invariant of $C…
Cubic forms over imaginary quadratic number fields and pairs of rational cubic forms
Christian Bernert, Leonhard Hochfilzer
We show that every cubic form with coefficients in an imaginary quadratic number field in at least variables represents zero non-trivially. This builds on the c…