Classification purely real algebraic irrationalities

N. M. Dobrovol'skii ; N. N. Dobrovol'skii ; D. K. Sobolev ; V. N. Soboleva

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 98-128.

Résumé

We study the appearance and properties of minimal residual fractions of polynomials in the decomposition of algebraic numbers into continued fractions. It is shown that for purely real algebraic irrationalities $\alpha$ of degree $n\ge2$, starting from some number $m_0=m_0(\alpha)$, the sequence of residual fractions $\alpha_m$ is a sequence of given algebraic irrationalities. The definition of the generalized number of Piso, which differs from the definition of numbers he's also the lack of any requirement of integrality. It is shown that for arbitrary real algebraic irrationals $\alpha$ of degree $n\ge2$, starting from some number $m_0=m_0(\alpha)$, the sequence of residual fractions $\alpha_m$ is a sequence of generalized numbers Piso. Found an asymptotic formula for the conjugate number to the residual fractions of generalized numbers Piso. From this formula it follows that associated to the residual fraction $\alpha_m$ are concentrated about fractions $-\frac{Q_{m-2}}{Q_{m-1}}$ is either in the interval of radius $O\left(\frac1{Q_{m-1}^2}\right)$ in the case of purely real algebraic irrationals, or in circles with the same radius in the General case of real algebraic irrationals, which have complex conjugate of a number. It is established that, starting from some number $m_0=m_0(\alpha)$, fair recurrent formula for incomplete private $q_m$ expansions of real algebraic irrationals $\alpha$, Express $q_m$ using the values of the minimal polynomial $f_{m-1}(x)$ for residual fractions $\alpha_{m-1}$ and its derivative at the point $q_{m-1}$. Found recursive formula for finding the minimal polynomials of the residual fractions using fractional-linear transformations.   Composition   this   fractional-linear transformation is a fractional-linear transformation that takes the system conjugate to an algebraic irrationality of $\alpha$ in the system of associated to the residual fraction, with a pronounced effect of concentration about rational fraction $-\frac{Q_{m-2}}{Q_{m-1}}$. It is established that the sequence of minimal polynomials for the residual fractions is a sequence of polynomials with equal discriminantly. In conclusion, the problem of the rational structure of a conjugate of the spectrum of a real algebraic irrational number $\alpha$ and its limit points. Bibliography: 28 titles.

Détail

Keywords: minimal polynomial, given an algebraic irrationality, generalized number Piso, residual fractions, continued fractions.

Bibliographie

[1] Aleksandrov A. G., “Computer investigation of continued fractions”, Algorithmic studies in combinatorics, Nauka, M., 1978, 142–161 ; 187 (Russian) | Zbl

[2] Berestovskii V. N., Nikonorov Yu. G., “Continued Fractions, the Group GL(2,Z), and Pisot Numbers”, Siberian Adv. Math., 17:4 (2007), 268–290 | DOI | MR | Zbl

[3] Bruno A. D., “Continued fraction expansion of algebraic numbers”, USSR Comput. Math. and Math. Phys., 4:2 (1964), 211–221 (Russian) | Zbl

[4] Bruno A. D., “Universal generalization of the continued fraction algorithm”, Chebyshevsky sbornik, 16:2 (2015), 35–65 (Russian)

[5] Weyl H., Algebraic Theory of Numbers, Annals of Mathematics Studies, 1, Princeton University Press, Princeton, N. J., 1940, viii+223 pp. ; Собр. соч. в 3-х томах, т. 1, Из-во АН УССР, Киев, 1952, 197–391 | MR | MR

[6] Voronoi G. F., On Generalization of the Algorithm of Continued Fraction, Warsawa University, 1896 | MR | Zbl

[7] Dobrovol'skii N. M., Hyperbolic Zeta function lattices, Dep. v VINITI 24.08.84, No 6090-84, 1984 (Russian)

[8] Dobrovol'skii N. M., Quadrature formulas for classes $E^\alpha_s(c)$ and $H^\alpha_s(c)$, Dep. v VINITI 24.08.84. No 6091-84, 1984 (Russian)

[9] Dobrovol'skii N. M., “About the modern problems of the theory of hyperbolic zeta-functions of lattices”, Chebyshevskii Sb., 16:1 (2015), 176–190 (Russian) | Zbl

[10] Dobrovol'skii N. M., Dobrovol'skii N. N., “About minimal polynomial residual fractions for algebraic irrationalities”, Chebyshevskii Sb., 16:3 (2015), 147–182 (Russian)

[11] N. M. Dobrovolskii, N. N. Dobrovolskii, I. N. Balaba, I. Yu. Rebrova, N. S. Polyakova, “Drobno-lineinye preobrazovaniya mnogochlenov i lineinye preobrazovaniya form”, Materialy XIII Mezhdunarodnoi konferentsii Algebra, teoriya chisel i diskretnaya geometriya: sovremennye problemy i prilozheniya, Dopolnitelnyi tom, Izd-vo Tul. gos. ped. un-ta im. L. N. Tolstogo, Tula, 2015, 134–149

[12] Dobrovol'skii N. M., Sobolev D. K., Soboleva V. N., “On the matrix decomposition of a reduced cubic irrational”, Chebyshevskii Sb., 14:1 (2013), 34–55 (Russian)

[13] Dobrovol'skii N. M., Yushina E. I., “On the reduction of algebraic irrationalities”, Algebra and Applications, Proceedings of the International Conference on Algebra, dedicated to the 100th anniversary of L. A. Kaloujnine (Nalchik, 6–11 September 2014), Publishing house KBSU, Nalchik, 2014, 44–46 (Russian)

[14] Dobrovol'skii N. M., Dobrovol'skii N. N., Yushina E. I., “On a matrix form of a theorem of Galois on purely periodic continued fractions”, Chebyshevskii Sb., 13:3 (2012), 47–52 (Russian) | Zbl

[15] Podsypanin V. D., “O razlozhenii irratsionalnostei chetvertoi stepeni v nepreryvnuyu drob”, Materialy mezhvuzovskoi nauchnoi konferentsii matematicheskikh kafedr pedinstitutov Tsentralnoi zony (Tula, 1968), 68–70

[16] Podsypanin V. D., “On the expansion of irrationalities of the fourth degree in the continued fraction”, Chebyshevskii Sb., 8:3(23) (2007), 43–46 (Russian) | Zbl

[17] Podsypanin E. V., “A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm”, Studies in number theory (LOMI) 4, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 67, 1977, 184–194 ; 227 (Russian) | Zbl

[18] Podsypanin E. V., “On the expansion of irrationalities of higher degrees in the generalized continued fraction (Materials V. D. Podsypanin) the manuscript of 1970”, Chebyshevskii Sb., 8:3(23) (2007), 47–49 (Russian) | Zbl

[19] Prasolov V. V., Polynomials, Translated from the 2001 Russian, Algorithms and Computation in Mathematics, 11, second edition by Dimitry Leites, Springer-Verlag, Berlin, 2004, xiv+301 pp. | MR | Zbl

[20] Trikolich E. V., Yushina E. I., “Continued fractions for quadratic irrationalities from the field $\mathbb Q(\sqrt 5)$”, Chebyshevskii Sb., 10:1 (2009), 77–94 (Russian)

[21] Feldman N. I., Priblizheniya algebraicheskikh chisel, Izd-vo Mosk. un-ta, M., 1981, 200 pp.

[22] Frolov K. K., “Upper bounds for the errors of quadrature formulae on classes of functions”, Dokl. Akad. Nauk SSSR, 231:4 (1976), 818–821 (Russian) | Zbl

[23] Frolov K. K., Kvadraturnye formuly na klassakh funktsiy, PhD thesis, VTS AN SSSR, M., 1979 (Russian)

[24] Yushina E. I., “About some the reduction of algebraic irrationalities”, Modern problems of mathematics, mechanics, Computer Science, Proceedings of the Regional scientific student conference, TulSU, Tula, 2015, 66–72 (Russian)

[25] E. I. Yushina, “O nekotorykh obobschennykh chislakh Pizo”, Universitet XXI veka: issledovaniya v ramkakh nauchnykh shkol, Materialy Vserossiiskoi nauchno-prakticheskoi konferentsii, TGPU im. L. N. Tolstogo, Tula, 2015, 66–72

[26] N. M. Dobrovol'skii, I. N. Balaba, I. Yu. Rebrova, N. N. Dobrovol'skii, “On Lagrange algorithm for reduced algebraic irrationalities”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2016, no. 2, 27–39 | MR | Zbl

[27] N. M. Dobrovol'skii, N. N. Dobrovolsky, I. N. Balaba, I. Yu. Rebrova, D. K. Sobolev, V. N. Soboleva, “Generalized Pisot Numbers and Matrix Decomposition”, Advances in Dynamical Systems and Control, Studies in Systems, Decision and Control, 69, eds. V. A. Sadovnichiy, M. Z. Zgurovsky, Springer International Publishing Switzerland, 2016 | DOI | MR | Zbl

[28] K. F. Roth, “Rational approximations to algebraic numbers”, Mathematika, 2 (1955), 1–20 ; corrigendum 168 | DOI | MR | Zbl