Quantitative subspace theorem and general form of second main theorem for higher degree polynomials

TL;DR

This paper advances Diophantine approximation and Nevanlinna theory by providing new bounds and generalizations for the subspace theorem and second main theorem involving higher degree polynomials and projective varieties.

Contribution

It introduces a new below bound for Chow weight and applies it to generalize the second main theorem and Schmidt's subspace theorem for higher degree polynomials.

Findings

New below bound for Chow weight of projective varieties

Quantitative Schmidt's subspace theorem for higher degree polynomials

General form of second main theorem for meromorphic mappings

Abstract

This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions.