论文标题
周期持续的分数在特征上是代数$ 2 $
Period-doubling Continued Fractions are Algebraic in Characteristic $2$
论文作者
论文摘要
考虑到一对独特的和非恒定的多项式，$ a $ a $ a $ a $ a $ a $ a $ \ inthbb {f} _2 [t] $，我们在$ \ mathbb {f} _2（（1/t）$中仅等于$ a $ a $ a $ a $ b $等于$ \ mathbb {f} _2（（1/t）$）中的持续分数。在第一作者和汉（出现在Acta arithmetica中）的先前作品中，作者考虑了两个案例，在这种情况下，部分商的顺序在每种情况下代表了著名且基本的2美元$ 2 $自动序列，这两种序列都以形态为类似的方式定义。他们可以在第一种情况下（Prouhet-thue-morse序列）证明相应的持续分数（a，b）$的相应持续分数的代数，并给出了第二种情况（周期级序列）的特定对证明。最近，Bugeaud和Han（Arxiv：2203.02213）在第一种情况下证明了任意对的代数。在这里，我们在第二种情况下为任意对提供了简短的证明。
Considering an arbitrary pair of distinct and non constant polynomials, $a$ and $b$ in $\mathbb{F}_2[t]$, we build a continued fraction in $\mathbb{F}_2((1/t))$ whose partial quotients are only equal to $a$ or $b$. In a previous work of the first author and Han (to appear in Acta Arithmetica), the authors considered two cases where the sequence of partial quotients represents in each case a famous and basic $2$-automatic sequence, both defined in a similar way by morphisms. They could prove the algebraicity of the corresponding continued fractions for several pairs $(a,b)$ in the first case (the Prouhet-Thue-Morse sequence) and gave the proof for a particular pair for the second case (the period-doubling sequence). Recently Bugeaud and Han (arXiv:2203.02213) proved the algebraicity for an arbitrary pair in the first case. Here we give a short proof for an arbitrary pair in the second case.
