Simultaneous approximation of modulus and values of Jacobi elliptic functions (in Ukrainian)

Author Ya. M. Kholyavka
Ivan Franko National University of L'viv

Abstract Let $\operatorname{sn}_i z$ be algebraically independent Jacobi elliptic functions, $(4K_i,2iK'_i)$ be main periods and $\varkappa_1, \varkappa_2$ be their moduli $\operatorname{sn}_i z$ ($i\in\{1,2\}$). We estimate from below the simultaneous approximation $\varkappa_1, \varkappa_2, \operatorname{sn}_1 K_2, \operatorname{sn}_2i K'_1$.
Keywords simultaneous approximation; Jacobi elliptic function
Reference 1. Bateman H., Erdelyi A. Higher transcendental functions. – M.: Nauka, V.3, 1967. (in Russian) 2. Lawden D.F. Elliptic functions and applications. – Springer-Verlag, Berlin, 1989. 3. Fel’dman N.I. Hilbert’s seventh problem. – M.: Moscov. Gos. Univ., 1982. (in Russian) 4. Fel’dman N.I., Nesterenko Yu.V. Transcendental Numbers. – Springer-Verlag, Berlin, 1998. 5. Reyssat E. Approximation algebrique de nombres lies aux fonctions elliptique et exp// Bull. Soc. Math. France. – 1980. – V.108. – P. 47–79. 6. Masser D. Elliptic functions and transcendence. – Springer-Verlag, Berlin, 1975. 7. Kholyavka Ya.M. On the simultaneous approximation of invariants of the elliptic function by algebraic numbers// Diophantine Analysis, Izd. Mosk. Gos. Univ., Moscow. – 1986. – Part 2. – P. 114–121. (in Russian) 8. Nesterenko Yu.V. On a measure of algebraic independence of values of an elliptic function// Izvestiya RAN: Ser. Mat. – 1995. – V.59, .4. – P. 155–178. (in Russian) 9. Chudnovsky G.V. Algebraic independence of the values of elliptic functions at algebraic points; Elliptic analogue of the Lindemann–Weierschtrass theorem// Inventiones Math. – 1980. – V.61. – P. 267–290.

Pages 10-15
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML