Systems of joint Thue polynomials for quadratic irrationalities

N. N. Dobrovol'skii ; N. M. Dobrovol'skii ; I. Yu. Rebrova ; E. A. Matveeva

Čebyševskij sbornik, Tome 23 (2022) no. 4, pp. 77-91

Résumé

The paper introduces a new concept — a system of joint Thue polynomials for a system of integer algebraic irrationalities. A parallel presentation of the elements of the theory of Thue polynomials for one algebraic irrationality and the foundations of the theory for a system of joint Thue polynomials for a system of integer algebraic irrationalities is carried out. A hypothesis is formulated about an analogue of the theorem of M. N. Dobrovolsky (Sr.) that for each order of $j$ there are two main Thue polynomials of the $j$th order, through which all the others are expressed. For a system of two quadratic irrationalities, for example, $\sqrt{2}$ and $\sqrt{3}$, systems of joint basic polynomials of order no lower than $0$, $1$ and $2$ are found. A theorem is proved on the general form of a pair of basic Thue polynomials of arbitrary order $n$ for quadratic irrationality $\sqrt{c}$, where $c$ is a square-free natural number.

Détail

Keywords: the minimum polynomial of the given algebraic irrationality, residual fractions, continued fractions, Tue pair, a system of joint Tue polynomials.