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，如果一个数是某个系数为整数的多项式的根，那么它就是代数的。数字x的次是任意整数多项式中x为根的最小次。众所周知，所有有理数的十进制表示最终都是周期的或有限的，因此1次代数数的情况是直接的。我们研究了一个关于高次代数数的类似问题。对这个问题的研究有着悠久的历史。它始于公元前300年左右古希腊欧几里得算法的发明。欧几里得算法最初是为计算两个整数的最大公约数而开发的。两千年后，欧几里得算法被用于二次无理数（即2次代数数）的研究。这里重要的一步是1695年J. Wallis引入正则连分式的概念。连分式将欧几里得算法与无理数联系在一起，特别是与二次无理数联系在一起。1770年j.l。拉格朗日证明了二次无理数的连分式的周期性，结束了二次情况下的问题（见第1节）。赫米特在1848年首次提出了将拉格朗日关于二次无理数连分式周期性的结果推广到三次代数数的问题。埃尔米特想知道是否存在三次无理性的周期性描述。对这个问题有许多不同的解释，这些解释导致了几何学和数论动力学的非凡理论。在这个项目中，我们将研究C. G. J. Jacobi于1868年提出的算法方法，并在1907年由O. Perron进一步发展。他们开发了一种多维连分式算法，称为雅可比-佩龙算法。Jacobi-Perron算法推广了欧几里得算法，并提供了类似于欧几里得算法提供的正则连分式的整数对序列。对于某些三次数，该算法的输出是周期性的，但是对于其他一些三次数，它被认为是非周期性的。由于这个原因，Jacobi-Perron算法并没有为Hermite问题提供一个完整的解决方案，但是它表明一种算法方法可能对这个问题有益。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伪以产生周期性输出。近年来，PI对Jacobi-Perron算法进行了两种新的改进：启发式代数周期性检测算法（简称启发式apd算法）和sin2算法。启发式apd算法在大量实验中证明了周期性，并被推测对所有三次数都是周期性的。sin2算法只适用于完全实数的情况（多项式的三个根都是实数）。对于sin2算法，我们能够证明三次共轭向量三元组的周期性。在全实三次情况下，sin2算法以Jacobi-Perron型算法的形式解决了Hermite问题。非完全实数的情况仍然开放，但是我们相信sin2算法的证明技术也可以适用于这种情况。这个项目的目的是继续对最后一个公开案例的周期性进行调查。现在正是解决这个问题、结束这个漫长故事的时候。

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.