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Third Italian Number Theory Meeting

Periodic representations and simultaneous approximations of cubic irrationalities

speaker: Nadir Murru (University of Turin, Department of mathematics)

abstract: In 1839 Hermite posed to Jacobi the problem of generalizing the construction of continued fractions to higher dimensions. In particular, he asked for a method of representing algebraic irrationalities by means of periodic sequences of integers that can highlight algebraic properties and possibly provide rational approximations. Hermite especially focused the attention on cubic irrationalities. Continued fractions completely solve this problem for quadratic irrationalities, but the problem for algebraic numbers of degree >2 is still open. Here, we present an approach to this problem developed in a recent paper of the author, where a periodic representation for all cubic irrationalities is provided. The periodic representation is provided by means of ternary continued fractions exploiting properties of linear recurrence sequences and generalizing classical R├ędei rational functions. The periodic representation is obtained starting from the knowledge of minimal polynomial of the involved cubic irrational. Thus, Hermite problem still remains open, since an algorithm defined over all real numbers is not defined. Moreover, we show rational approximations that arise from the periodic representation by means of powers of 3X3 matrices. These approximations can be generalized by means of regular representation of cubic extensions. In this way, we found a generalization of Khovanskii matrices (which are only used for approximations of cubic roots) that allows to provide rational approximations given any cubic irrational. The method produce rational approximations in a fast way with more accuracy (considering same size of denominators) than approximations provided by iterative methods, as Newton and Hallley ones.

Mon 21 Sep, 15:30 - 15:55, Aula Russo
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