论文标题
$ p $的收敛条件 - ADIC持续分数
Convergence conditions for $p$--adic continued fractions
论文作者
论文摘要
几位作者在$ p $ $ p $ $ \ mathbb {q} _p $的字段中引入了持续的分数。但是，由于所有提出的算法无法复制$ \ Mathbb {r} $中的所有续分的所有属性，因此仍然缺少标准定义。特别是，尚未证明对Lagrange定理的类似物，因为将持续分数概括为$ \ Mathbb {q} _p $。因此，值得研究$ p $的新算法的定义 - ADIC持续分数。新方法需要实现的主要条件是持续分数的$ \ mathbb q_p $中的收敛性。在本文中，我们研究了$ \ mathbb {q} _p $中持续分数的一些收敛条件。这些结果允许定义许多持续分数的新家族，它们可以保证融合。然后，我们提供一些利用新收敛条件的新算法，并证明其中一个在输入是理性的时终止了有限数量的步骤，因为它发生在实际持续的分数。
Continued fractions have been introduced in the field of $p$--adic numbers $\mathbb{Q}_p$ by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in $\mathbb{R}$. In particular, an analogue of the Lagrange's Theorem is not yet proved for any attempt of generalizing continued fractions in $\mathbb{Q}_p$. Thus, it is worth to study the definition of new algorithms for $p$--adic continued fractions. The main condition that a new method needs to fulfill is the convergence in $\mathbb Q_p$ of the continued fractions. In this paper we study some convergence conditions for continued fractions in $\mathbb{Q}_p$. These results allow to define many new families of continued fractions whose convergence is guaranteed. Then we provide some new algorithms exploiting the new convergence condition and we prove that one of them terminates in a finite number of steps when the input is rational, as it happens for real continued fractions.
