发表机构
Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对三次Thue--Mahler方程证明了对regulator具节能依赖的有效界，将Mordell方程整数解的Stark界指数降至1/2+ε，改进了相关分离下界并在Frey高度猜想的导体方面取得节能进展。
AI 中文摘要
我们针对三次Thue--Mahler方程证明了新的有效界，该界对 regulator 具有节能依赖关系，此成果有若干应用。首先，我们将Mordell方程$y^2=x^3+k$（$k$为非零整数）整数解的Stark界$\log \max\{|x|,|y|\} \ll_\epsilon |k|^{1+\epsilon}$的指数$1+\epsilon$降至$1/2+\epsilon$，这是50多年来首次对$k$无限制的节能改进。其次，针对大小为$O(T)$的整数平方与立方，我们将Stark在1973年得到的已知无条件分离下界$(\log T)^{1-o(1)}$改进为$(\log T)^{2-o(1)}$。最后，对于$\mathbb{Q}$上具有整$j$-不变量的椭圆曲线，我们在Frey高度猜想（Szpiro猜想的强化形式）现有最强界的导体方面取得了节能改进。
英文摘要
We prove a new effective bound for cubic Thue--Mahler equations with power-saving dependence on the regulator. This has some applications. First, we improve Stark's bound $\log \max\{|x|,|y|\} \ll_ε|k|^{1+ε}$ for the integer solutions of Mordell's equation $y^2=x^3+k$ ($k$ a non-zero integer) by reducing the exponent $1+ε$ to $1/2+ε$; this is the first power-saving improvement without restrictions on $k$ in more than 50 years. Secondly, for integer squares and cubes of size $\asymp T$ we improve the known unconditional separation lower bound $(\log T)^{1-o(1)}$ obtained by Stark in 1973 to $(\log T)^{2-o(1)}$. Finally, we obtain a power-saving improvement in the conductor aspect of the strongest currently available bounds for Frey's height conjecture (a strengthening of Szpiro's conjecture) in the case of elliptic curves over $\mathbb{Q}$ with integral $j$-invariant.
