On the p-adic Second Main Theorem

TL;DR

This paper improves the non-archimedean Second Main Theorem in Nevanlinna theory, providing a sharper defect relation for analytic maps intersecting hypersurfaces with certain conditions.

Contribution

It offers an improved version of the Second Main Theorem for non-archimedean analytic maps with sharper defect bounds under transversality.

Findings

Established a sharper defect relation _f(D_i)  n-1+1/d for non-archimedean maps.

Proved the defect relation is sharp for all positive integers n and d.

Extended the Second Main Theorem to cases with hypersurfaces of degree greater than one.

Abstract

We study the Second Main Theorem in non-archimedean Nevanlinna theory, giving an improvement to the non-archimedean Second Main Theorems of Ru and An in the case where all the hypersurfaces have degree greater than one and all intersections are transverse. In particular, under a transversality assumption, if f is a nonconstant non-archimedean analytic map to P^n and D_1,..,D_q are hypersurfaces of degree d, we prove the defect relation \sum_{i=1}^q\delta_f(D_i)\leq n-1+1/d, which is sharp for all positive integers n and d.