Periodicity of Jacobi-Perron type algorithms for cubic vectors.

这个项目致力于研究代数数的周期表示。回想一下，如果一个数是某个整数系数多项式的根，那么它就是代数数。一个数x的次数是任何整数多项式的最小次数，其中x是根。众所周知，所有有理数的十进制表示最终都是周期性的或有限的，因此1次代数数的情况很简单。我们研究一个类似的问题，代数数的更高的程度。对这个问题的研究有着丰富的历史。它开始于古希腊，大约在公元前300年发明了欧几里得的算法。欧几里得算法最初是为了计算两个整数的最大公约数而开发的。两千年后的欧几里德算法被用于研究二次无理数（即代数数的程度2）。一个重要的一步，这里是介绍的概念，经常继续分数的J.沃利斯在1695年。连分数将欧几里得的算法与无理数联系起来，特别是与二次无理数联系起来。在1770年J. - L.拉格朗日证明了周期性的连续分数的二次无理数，关闭问题的二次情况下（见第1节）。赫密特首先提出的问题generalising拉格朗日的结果的周期性继续分数的二次无理数的情况下代数数的程度三在1848年。埃尔米特想知道是否有一个周期性的描述立方无理数。有许多不同的解释这个问题，导致显着的理论，几何和动力学的数字。在这个项目中，我们将研究由C。G. J. Jacobi在1868年提出的，并由O. 1907年的Perron。他们开发了一种多维连分数算法，称为Jacobi-Perron算法。Jacobi-Perron算法推广了欧几里德算法，并提供了一个类似于欧几里德算法提供的正则连分数的整数对序列。该算法的输出对于某些立方数是周期性的，但是对于其他一些立方数，它被认为是非周期性的。由于这个原因，雅可比-佩龙算法没有提供一个完整的解决方案，厄米的问题，但它表明，算法的方法可能是有益的问题。类似的情况发生在过去100年中引入的许多其他Jacobi-Perron类型算法中，这些算法既没有被证明也没有被证明产生周期性输出。最近PI提出了两种新的Jacobi-Perron算法的改进：启发式代数周期检测算法（或简称启发式APD算法）和sin 2算法。启发式APD算法在大量的实验中证明了周期性，并被证明对所有立方数都是周期性的。sin 2-算法只在全真实的情况下工作（多项式的所有三个根都是真实的数）。对于sin 2算法，我们能够证明三次共轭向量的周期性。sin 2-算法以Jacobi-Perron型算法的形式对全真实的三次问题提供了一个解答。非完全真实的情况下仍然开放，但我们相信，证明的技术sin 2算法可以适用于这种情况。该项目的目的是继续调查上一个未决案件的周期性。现在是解决这个问题并结束这个漫长故事的正确时机。

This project is dedicated to studying the periodic representations of algebraic numbers. Recall that a number is algebraic if it is a root of some polynomial with integer coefficients. The degree of a number x is the smallest degree of any integer polynomial for which x is a root. It is well known that decimal representations of all rational numbers are eventually periodic or finite, so the case of algebraic numbers of degree 1 is straightforward. We study a similar question for algebraic numbers of higher degrees. The study of this question has a rich history. It begins in ancient Greece with the invention of Euclid's algorithm around 300 BC. Euclid's algorithm was originally developed for computing the greatest common divisor of two integers. Two millennia later the Euclidean algorithm was being used in the study of quadratic irrationals (i.e. algebraic numbers of degree 2). An important step here was the introduction of the concept of regular continued fractions by J. Wallis in 1695. Continued fractions link Euclid's algorithm to irrational numbers in general and to quadratic irrationalities in particular. In 1770 J.-L. Lagrange proved the periodicity of continued fractions for quadratic irrationalities, closing the question for the quadratic case (see Section 1). Ch. Hermite first posed the problem of generalising Lagrange's result on the periodicity of continued fractions for quadratic irrationalities to the case of algebraic numbers of degree three in 1848. Hermite wondered if there is a periodic description of cubic irrationalities. There are many different interpretations of this question that led to remarkable theories in geometry and dynamics of numbers. For this project we will study the algorithmic approach to the problem that was initiated by C. G. J. Jacobi in 1868 and further developed by O. Perron in 1907. They developed a multidimensional continued fraction algorithm, known as the Jacobi-Perron algorithm. The Jacobi-Perron algorithm generalises the Euclidean algorithm and provides a sequence of pairs of integers similar to the regular continued fractions provided by the Euclidean algorithm. The output of the algorithm is periodic for certain cubic numbers, however it is believed to be non-periodic for some others. For that reason the Jacobi-Perron algorithm does not provide a complete solution to Hermite's problem, however it suggests that an algorithmic approach might be beneficial to the question. A similar situation occurs with numerous other Jacobi-Perron type algorithms introduced in the last 100 years, that are neither proved nor disproved to produce a periodic output. Recently PI have introduced two new modifications of the Jacobi-Perron algorithm: the heuristic algebraic periodicity detecting algorithm (or heuristic APD-algorithm for short) and sin2-algorithm. The heuristic APD-algorithm demonstrates periodicity in numerous experiments and is conjectured to be periodic for all cubic numbers. The sin2-algorithm works only in the totally real case (all three roots of the polynomial are real numbers). For the sin2-algorithm we were able to prove periodicity for triples of cubic conjugate vectors. The sin2-algorithm provides an answer to Hermite's problem in the form of Jacobi-Perron type algorithm for the totally real cubic case. The non-totally-real case remains open, however we believe that the techniques of the proof for the sin2-algorithm can be adapted for that case as well. The aim of this project is to continue the investigation of periodicity in the last open case. It is a right time to attack this problem and put the end to this long story.