[Paper Review] On the Hermite problem for cubic irrationalities

Nadir Murru|arXiv (Cornell University)|May 14, 2013

Advanced Mathematical Identities16 references3 citations

TL;DR

This paper solves the Hermite problem for cubic irrationalities by constructing a periodic ternary continued fraction with rational partial quotients that converges to any cubic irrationality. The method modifies the Jacobi algorithm to produce a pre-period of length 2 and a period of length 3, providing simultaneous rational approximations and proving periodicity via algebraic properties of cubic irrationals and linear recurrence sequences.

ABSTRACT

In this paper, the Hermite problem has been approached finding a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In other words, the problem of writing cubic irrationals as a periodic sequence of rational or integer numbers has been solved. In particular, a periodic multidimensional 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. This multidimensional continued fraction is derived from a modification of the Jacobi algorithm, which is proved periodic if and only if the inputs are cubic irrationals. Moreover, this representation provides simultaneous rational approximations for cubic irrationals.

Motivation & Objective

• To address the open problem of finding a periodic representation for cubic irrationalities using multidimensional continued fractions.
• To prove that any cubic irrationality can be expressed as a periodic sequence of rational or integer numbers via a modified Jacobi algorithm.
• To establish a direct construction of such a periodic representation using algebraic properties of cubic irrationals and linear recurrent sequences.
• To provide simultaneous rational approximations for cubic irrationals through the new continued fraction framework.

Proposed method

• A modified Jacobi algorithm is introduced, generating a ternary continued fraction with pre-period of length 2 and period of length 3.
• The construction uses the algebraic identity α³ = pα² + qα + r for cubic irrationals α, and manipulates expressions involving r/α and α to derive recurrence relations.
• The method applies functions fₙ^α and gₙ^α to transform variables and extract integer parts, ensuring periodicity through trace and determinant identities of a 2×2 matrix N derived from the cubic equation.
• The convergence of the continued fraction to the pair (r/α, α) is proven using properties of linear recurrent sequences and the characteristic polynomial of matrix N.
• The algorithm is shown to become periodic when the input is a cubic irrationality, with the periodic structure arising from the algebraic invariants of the cubic field.
• Cerruti polynomials are introduced as a generalization of Rédei rational functions, providing a structural link to number-theoretic applications.

Experimental results

Research questions

• RQ1 Can a periodic representation of cubic irrationalities be constructed using a multidimensional continued fraction with rational or integer partial quotients?
• RQ2 Does the modified Jacobi algorithm produce a periodic sequence when applied to any cubic irrationality?
• RQ3 Can the ternary continued fraction provide simultaneous rational approximations to cubic irrationals?
• RQ4 What is the role of the parameter z in determining the periodicity and convergence rate of the continued fraction?
• RQ5 Can the convergence of the continued fraction be linked to linear recurrence sequences in the numerators and denominators of its convergents?

Key findings

• A periodic ternary continued fraction with pre-period length 2 and period length 3 is constructed that converges to any cubic irrationality α satisfying x³ - px² - qx - r = 0.
• The representation is explicitly derived using algebraic identities and matrix invariants, with the periodicity arising from the trace and determinant of a 2×2 matrix N associated with the cubic field.
• The algorithm is proven to be periodic if and only if the input is a cubic irrationality, establishing a necessary and sufficient condition for periodicity.
• Simultaneous rational approximations are provided by the convergents of the continued fraction, which are shown to be linear recurrent sequences.
• The method yields a direct construction of periodic representations without relying on iterative convergence, solving the Hermite problem for cubic irrationals in a constructive way.
• Cerruti polynomials are identified as a generalization of Rédei functions and are shown to play a key role in the periodic structure of the continued fraction.