A $p$-adic Second Main Theorem

TL;DR

This paper extends the Second Main Theorem in value distribution theory to non-archimedean fields, providing new estimates for analytic maps intersecting hypersurfaces in projective space.

Contribution

It generalizes Levin's work by establishing a Second Main Theorem estimate for hypersurfaces of arbitrary degree in non-archimedean settings.

Findings

Established a Second Main Theorem estimate for analytic maps in non-archimedean fields.

Extended previous results to hypersurfaces not all being hyperplanes.

Provides a framework for further research in non-archimedean value distribution theory.

Abstract

Let $\mathbf{K}$ be an algebraically closed field of arbitrary characteristic, complete with respect to a non-archimedean absolute value $|\,|$. We establish a Second Main Theorem type estimate for analytic map $f\colon \mathbf{K}\rightarrow\mathbb{P}^n(\mathbf{K})$ and a family of $n$ hypersurfaces in $\mathbb{P}^n(\mathbf{K})$ intersecting transversally and not all being hyperplanes. This implements the previous work of Levin where the case of all hypersurfaces having degree greater than one was studied.