Pencils of norm form equations and a conjecture of Thomas, II
Abstract
We continue our studies on parametric norm forms Ft(bf x), with bf x=(x0,x1,…,xd−1) lying in some parametric linear subvariety Wt and integers t sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions bf x of Ft(bf x)=1. Here we modify our techniques to treat Ft(bf x)=q for an arbitrary integer q. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all bf x are polynomially bounded in terms of |q| and t. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with x2=⋯=xd−1=0. We present an example with eight variables.
Related
- Adjoint Closures of Singular Quadratic Pencils and First's Pfister-Type Conjecture
- Asymptotic Neutrality for the Ionization Conjecture in the Two-Center Thomas-Fermi-Dirac-von Weizsäcker Theory
- Complete self-shrinkers with constant norm of the second fundamental form in RR4
- Rigidity of Three-Dimensional Self-Shrinkers with Constant Norm of the Second Fundamental Form
- The second gap for self-shrinkers with constant norm of the second fundamental form
BibTeX
@misc{amoroso2026pencils,
title = {{Pencils of norm form equations and a conjecture of Thomas, II}},
author = {Francesco Amoroso and David Masser and Umberto Zannier},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09995},
eprint = {2609.09995},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09995},
note = {CC0},
}
RIS
TY - UNPB
TI - Pencils of norm form equations and a conjecture of Thomas, II
AU - Francesco Amoroso
AU - David Masser
AU - Umberto Zannier
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09995
UR - https://arxiv.org/abs/2609.09995
AB - We continue our studies on parametric norm forms F_t(bf x), with bf x=(x_0,x_1,…,x_d−1) lying in some parametric linear subvariety W_t and integers t sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions bf x of F_t(bf x)=1. Here we modify our techniques to treat F_t(bf x)=q for an arbitrary integer q. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all bf x are polynomially bounded in terms of |q| and t. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with x_2=⋯=x_d−1=0. We present an example with eight variables.
ER -