Explicit and Mixed Estimates for Thue inequalities with few coefficients

TL;DR

The paper delivers fully explicit upper bounds for the number of integer solutions to the Thue inequality $|F(x,y)|\le h$ for irreducible forms $F$ of degree $r\ge 3$ with at most $s+1$ nonzero coefficients. Building on Thue–Siegel type principles, Schmidt clustering, and Bombieri–Schmidt methods, it develops three explicit counting bounds (Main1–Main3) by combining primitive-solution reductions, discriminant amplification via $M_2(\mathbb{Z})^{\times}$ actions, height/Mahler reduction, and Diophantine-approximation tools. The results give concrete constants in terms of $r$, $s$, $h$, and $|D(F)|$, and cover large, small, and intermediate solution regimes through a unified framework, improving prior bounds by making the implicit constants explicit. A careful correction to Bengoechea’s theorem finalizes the discussion, ensuring the validity of the explicit bounds across discriminant ranges. The work provides practically computable criteria for solution counts and advances the explicit-constant theory for Thue inequalities with few coefficients.

Abstract

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were written giving an upper bound for the number of solutions of the above inequality as $\ll c(r,s,h)$ where $c(r,s,h)$ is an explicit function of $r,s$ and $h.$ Invariably, the absolute constant involved in $\ll$ has been left undetermined. In this paper, following Bombieri, Schmidt and Mueller, we give three different upper bounds which are explicit in every aspect.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• Corollary 3.3
• Lemma 3.4
• proof