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Chebyshevskii Sbornik, 2022, Volume 23, Issue 4, Pages 77–91
DOI: https://doi.org/10.22405/2226-8383-2022-23-4-77-91
(Mi cheb1224)
 

Systems of joint Thue polynomials for quadratic irrationalities

N. N. Dobrovol'skiiab, N. M. Dobrovol'skiia, I. Yu. Rebrovaa, E. A. Matveevac

a Tula State Lev Tolstoy Pedagogical University (Tula)
b Tula State University (Tula)
c Center for Creative Development and Humanitarian Education (Suvorov, Tula Region)
References:
Abstract: 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.
Keywords: the minimum polynomial of the given algebraic irrationality, residual fractions, continued fractions, Tue pair, a system of joint Tue polynomials.
Funding agency Grant number
Russian Foundation for Basic Research 19-41-710004_р_а
The research was carried out with the financial support of the Russian Foundation for Basic Research within the framework of a scientific project No.19-41-710004_p_a and with the financial support of a grant from the Government of the Tula region under the Agreement DS/294 dated 16.11.2021.
Received: 17.06.2022
Accepted: 08.12.2022
Document Type: Article
UDC: 511.3
Language: Russian
Citation: N. N. Dobrovol'skii, N. M. Dobrovol'skii, I. Yu. Rebrova, E. A. Matveeva, “Systems of joint Thue polynomials for quadratic irrationalities”, Chebyshevskii Sb., 23:4 (2022), 77–91
Citation in format AMSBIB
\Bibitem{DobDobReb22}
\by N.~N.~Dobrovol'skii, N.~M.~Dobrovol'skii, I.~Yu.~Rebrova, E.~A.~Matveeva
\paper Systems of joint Thue polynomials for quadratic irrationalities
\jour Chebyshevskii Sb.
\yr 2022
\vol 23
\issue 4
\pages 77--91
\mathnet{http://mi.mathnet.ru/cheb1224}
\crossref{https://doi.org/10.22405/2226-8383-2022-23-4-77-91}
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