求解特征为$(p,p,2)$的方程系数在数域上
Solving equations of signature $(p,p,2)$ with coefficients over number fields
AI总结：
本文利用模方法研究了特征为$(p,p,2)$的数域上丢番图方程的解，并给出了特定数域中非平凡解的显式界。
摘要：
利用模方法，我们研究了数域上丢番图方程$$Aa^p+Bb^p=Cc^2$$的解。我们首先证明了在满足适当$S$-单位条件的一般数域中，一个渐近结果，假定在非完全实数域的情况下某些标准猜想成立。具体来说，我们验证了这个条件对无限多个实二次域成立。在非渐近设置下，我们也获得了有效结果。特别是对于方程$$a^p+db^p=c^2$$在$K= \mathbb{Q}(\sqrt{-d})$上，其中$d \in \{3, 11, 19, 43, \}$和$K= \mathbb{Q}(\sqrt d)$上，其中$d \in \{3, 5, 11, 13, 19, 29\}$，我们找到了依赖于$d$的显式界，使得当$p$超过这些界时，不存在某种类型的非平凡解。
英文摘要：
Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some standard conjectures in the case of fields that are not totally real. Specifically, we verify that this condition holds for an infinite family of real quadratic fields. Outside the asymptotic setting, we also obtain effective results. In particular, for the equation $$a^p+db^p=c^2$$ over $K= \mathbb{Q}(\sqrt{-d})$ with $d \in \{3, 11, 19, 43, \}$ and $K= \mathbb{Q}(\sqrt d)$ with $d \in \{3, 5, 11, 13, 19, 29\}$, we find explicit bounds (depending on $d$) such that no non-trivial solutions of a certain type exist whenever $p$ exceeds these bounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
