论文标题
在固定数字字段中具有多个指数的扭曲thue方程
Twisted Thue equations with multiple exponents in fixed number fields
论文作者
论文摘要
令$ k $为$ d \ geq 3 $的数字字段，并修复$ s $乘法性地倍增独立的代数整数$γ_1，\ dots，γ_s\ in k^*$，可以满足某些技术要求，可以将其简化为$ \ \ \ \ m athbb {q} $ - {q} $ - 线性的独立性，允许使用Schanuel的独立，允许使用Schanuel的命令。然后，我们考虑扭曲的thue方程\ [ \ left | n_ {k/\ mathbb {q}}} \ left（x-γ_1^{t_1} \cdotsγ_S^{t_s} y \ right）\ right | = 1，\]，并证明它只有许多解决方案$（x，y，（t_1，\ dots，t_s））$，带有$ xy \ neq 0 $和$ \ mathbb {q} \ left（γ_1^{γ_1^{t_1} \ cdotsγ_Sγ_Sγ_Sγ_Sγ_S^{
Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_1, \dots, γ_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly independence, given Schanuel's conjecture. We then consider the twisted Thue equation \[ \left|N_{K/\mathbb{Q}}\left(X-γ_1^{t_1}\cdotsγ_s^{t_s}Y\right)\right| = 1, \] and prove that it has only finitely many solutions $(x,y, (t_1, \dots, t_s) )$ with $xy \neq 0$ and $\mathbb{Q}\left( γ_1^{t_1}\cdots γ_s^{t_s} \right) = K$, all of which are effectively computable.
