Diophantine Approximation of non-algebraic points on varieties II: Explicit estimates for arithmetic Hilbert Functions

TL;DR

This paper provides explicit estimates for arithmetic Hilbert functions of subvarieties in projective space to address the ineffectiveness of the traditional arithmetic Hilbert-Samuel formula in Diophantine approximation.

Contribution

It introduces explicit bounds for arithmetic Hilbert functions, enabling more effective Diophantine approximation on varieties.

Findings

Derived explicit estimates for arithmetic Hilbert functions.

Improved methods for Diophantine approximation on algebraic varieties.

Addressed the ineffectiveness of classical formulas in this context.

Abstract

Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic Hilbert Functions of closed subvarieties in projective space.