Tubular approaches to Baker's method for curves and varieties

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

Upper bounds for the number of solutions to quartic Thue equations

We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve (x,y) that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of author's earlier work with Okazaki [2] by giving special treatments to forms with respect to their signature

Extremal families of cubic Thue equations

We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear forms in logarithms and lattice-basis reduction