On the periodic writing of cubic irrationals and a generalization of Rédei functions

Abstract

In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomal with rational coefficients, we study the Cerruti polynomials µUsing these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensioanl continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

On a periodic Jacobi-Perron type algorithm

“…This concludes the proof of Lemma 8. 19. (Note that all the estimations of this proof checked directly by symbolic computations in MAPLE2020, see Case-2.mw in [11]).…”

Approximations of Algebraic Irrationalities with Matrices

“…Finally, let α be the real root largest in modulus of t 3 −pt 2 −qt−r, with p, q, r ∈ Q. In [12], the author showed that matrix…”

“…yields simultaneous rational approximations of α − p and r α . However, in [12] the author did not focus on the study of rational approximations, but studied matrix C in order to determine periodic representations for any cubic irrational.…”

Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

“…The periodicity of the Jacobi-Perron algorithm is also related to the study of Pisot numbers [19,20]. Further studies on MCFs can be found in [2,16,29,41]. Continued fractions for p-adic numbers were introduced in three different ways [10,36,37] and subsequently studied by several authors like [7,11,13,21].…”