On Hermite’s problem, Jacobi–Perron type algorithms, and Dirichlet groups
A well-known result of Lagrange (1770) characterises quadratic irrationalities as those real numbers that can be written as periodic continued fractions. Hermite asked in 1848 if there exists some way to write cubic irrationalities periodically. To approach this problem, Jacobi and Perron generalised the classical continued fraction algorithm to the three-dimensional case; this algorithm is now called the Jacobi–Perron algorithm. It is known only to provide periodicity for some cubic irrationalities.
In this paper we introduce two new algorithms in the spirit of the Jacobi–Perron algorithm: the heuristic algebraic periodicity detecting algorithm and the $\sin ^2$-algorithm. The heuristic algebraic periodicity detecting algorithm is a very fast and efficient algorithm, and its output is periodic for numerous examples of cubic irrationalities, but its periodicity for cubic irrationalities is not proven. The $\sin ^2$-algorithm is limited to the totally-real cubic case (where all the roots of cubic polynomials are real numbers). Recently we proved the periodicity of the $\sin ^2$-algorithm for all cubic totally-real irrationalities. To the best of our knowledge this is the first Jacobi–Perron type algorithm for which the cubic periodicity is proven. The $\sin ^2$-algorithm provides an answer to Hermite’s problem for the totally real case (the case of cubic algebraic numbers with complex conjugate roots remains open).
We conclude this paper with an application of Jacobi–Perron type algorithms to the computation of independent elements in the maximal groups of commuting matrices of algebraic irrationalities.