# Tubular approaches to Baker's method for curves and varieties

## TL;DR

This paper generalizes Baker's method for varieties, combining it with Runge's method to improve estimates for integral points on curves, reducing reliance on complex logarithmic forms.

## Contribution

It introduces a novel approach that merges Baker's and Runge's methods, enhancing effectiveness in bounding integral points on algebraic varieties.

## Key findings

- Generalization of Baker's method for varieties.
- Combination of methods improves estimates for curves.
- Reduction in reliance on p-adic logarithms.

## Abstract

Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties. In this article, we give a generalisation of results of Levin regarding Baker's method for varieties, and explain how, quite surprisingly, it mixes (under additional hypotheses) with Runge's method to improve some known estimates in the case of curves by bypassing (or more generally reducing) the need for linear forms in $p$-adic logarithms.