Periodicity of Jacobi-Perron type algorithms for cubic vectors.

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.