Contributions to a conjecture of Mueller and Schmidt on Thue inequalities
N. Saradha, Divyum Sharma

TL;DR
This paper improves bounds on the number of solutions to Thue inequalities for certain forms, reducing the dependence from s^2 to a smaller function involving s, logs, and exponential terms, under specific coefficient conditions.
Contribution
It refines the upper bounds on solutions to Thue inequalities, replacing s^2 with more precise functions based on form parameters and coefficient conditions.
Findings
Bound s^2 replaced by max(s log^3 s, s e^{ ext{ extPsi}})
Bound s^2 replaced by s log^{3/2} s under symmetry conditions
Improves understanding of solution counts for forms with specific coefficient structures
Abstract
Let be a form of degree , irreducible over and having at most non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality \[ |F(X,Y)|\leq h \] is . They that may be replaced by . Let \[ \Psi = \max_{0\leq i\leq s} \max\left( \sum_{w=0}^{i-1}\frac{1}{r_i-r_w},\sum_{w= i+1}^{s}\frac{1}{r_w-r_i}\right). \] Then we show that may be replaced by . We also show that if and for , then may be replaced by . In particular, this is true if .
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Taxonomy
TopicsAnalytic Number Theory Research
