Systems of joint Thue polynomials for quadratic irrationalities
https://doi.org/10.22405/2226-8383-2022-23-4-77-91
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 𝑗th order, through which all the others are expressed. For a system of two quadratic irrationalities, for example, √2 and √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 √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.
Supporting agencies: 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.
About the Authors
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University; Tula State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Nikolai Mihailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Elizabeth Alexandrovna Matveeva
Center for Creative Development and Humanitarian Education
Russian Federation
Russian Federation
References
1. Dobrovol’skii M. N. 2010 "On the decomposition of irrationalities of the third degree into continuous fractions" Chebyshevsky Sb. vol. XI, №. 4 (36). pp. 4–24.
2. V. A. Kretschmar 1939, "On the upper limit of the number of representations of an integer by some binary forms of the fourth degree" , Izv. AN USSR. Ser. matem. Vol. 3, issue 3. pp. 289–302.
3. E. A. Morozova 2015, "Tue polynomials for quadratic irrationalities" , Algebra, number theory and discrete geometry: modern problems and applications : Proceedings of the XIII International Conference, Tula, April 15-17, 2015 / Tolstoy Tula State Pedagogical University. – Tula: Tula State Pedagogical University named after L.N. Tolstoy, – pp. 161-168.
4. Podsypanin V. D. 2010 "On Thue polynomials and the expansion of irrationalities of the fourth degree into a continued fraction"Chebyshevskii sbornik. T. XI, vol. 4 (36). pp. 25–69.
5. Siegel C. L. 1929, "¨Uber einige Anwendungen Diophantischer Approximationen" , Abhandlungen der Preuss. Akad. d. Wissensch., Phys.-Math. Klasse. PP. 1–70.
6. Thue A. 1910. "¨Uber Ann¨aherungswerte algebraischer Zahlen" , J. reine ang. Math. Vol. 135. PP. 284–305.
Review
For citations:
Dobrovol’skii N.N., Dobrovol’skii N.M., Rebrova I.Yu., Matveeva E.A. Systems of joint Thue polynomials for quadratic irrationalities. Chebyshevskii Sbornik. 2022;23(4):77-91. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-4-77-91
Views:
253
Email this article 