AI 中文总结
本文在特征零的域上，改进拟周期连分数的Maillet-Baker判据，证明函数域内山定理的定量版本，给出代数幂级数异常好有理逼近的明确界，还得到收敛分母增长的上界，且证明回文连分数的二分性。
AI 中文摘要
给定域$K$，设$K((T^{-1}))$为形式幂级数域。$K((T^{-1}))$中的连分数可通过类比经典实连分数定义，且已被广泛研究。已有部分结果确立了由特殊连分数族得到的$K((T^{-1}))$中元素的超越性，但仍有诸多待探索内容。本文假设$K$的特征为零，改进了拟周期连分数的Maillet-Baker判据的已知类似结果。我们证明的核心工具是函数域内山定理（Roth定理的函数域类似）的定量版本，该版本为代数幂级数的异常好有理逼近的数量给出了明确界。此定量估计还给出了代数元素的收敛分母增长的Davenport-Roth型上界。最后，我们证明回文连分数要么是二次的，要么是超越的，与实情形一致，但采用了不同的证明策略。
英文摘要
Given a field $K$, let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.
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