An isoperimetric inequality related to Thue's equation

Cubic Thue inequalities with positive discriminant

We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided that h is smaller than D^{1/4}.

Diagonalizable Thue Equations -- revisited

Let $r,h\in\mathbb{N}$ with $r\geq 7$ and let $F(x,y)\in \mathbb{Z}[x ,y]$ be a binary form such that \[ F(x , y) =(αx + βy)^r -(γx + δy)^r, \] where $α$, $β$, $γ$ and $δ$ are algebraic constants with $αδ-βγ\neq 0$. We establish upper bounds for the number of primitive solutions to the Thue inequality $0<|F(x, y)| \leq h$, improving an earlier result of Siegel and of Akhtari, Saradha & Sharma.

少系数Thue不等式的显式与混合估计

Explicit and Mixed Estimates for Thue inequalities with few coefficients

本文针对少系数Thue不等式，在Bombieri、Schmidt和Mueller方法的基础上，给出了三个完全显式的解数上界，解决了此前绝对常数未定的问题。