On estimates for polynomials in values of $E$-functions

A. B. Shidlovskii

Sbornik. Mathematics, Tome 43 (1982) no. 1, pp. 1-32

Résumé

In this article the notions of measures of linear independence, transcendence and relative transcendence of numbers are generalized to the case when the linear forms and polynomials in the definition have algebraic coefficients. An axiomatization is given for a method of estimating the measures from the values at algebraic points of a set of $E$-functions satisfying linear differential equations with coefficients in the field of rational functions. Several theorems which estimate such measures are proved in the case when the coefficients of the power series of the $E$-functions under consideration, the coefficients of the polynomials in the measures and the values of the argument belong to an arbitrary algebraic number field. Here most of the theorems proved relate to the case when one estimates the measures of a subset of values of $E$-functions, and the basic set of $E$-functions being considered is algebraically dependent over the field of rational functions. As corollaries of these estimates, some important arithmetic properties of the values of sets of functions which are products of powers of these functions are obtained. Bibliography: 31 titles.

Bibliographie

[1] Siegel C. L., “Über einige Anwendungen Diophantischer Approximationen”, Abhand. preuss. Acad. Wiss., 1929–1930, no. 1, 1–70 | Zbl

[2] Siegel C. L., Transcendental numbers, Princeton University press, 1949 | MR

[3] Shidlovskii A. B., “O kriterii algebraicheskoi nezavisimosti znachenii odnogo klassa tselykh funktsii”, Izv. AN SSSR, ser. matem., 23 (1959), 35–66 | MR | Zbl

[4] Shidlovskii A. B., “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii $E$-funktsii, svyazannykh lyubym chislom algebraicheskikh uravnenii v pole ratsionalnykh funktsii”, Izv. AN SSSR, ser. matem., 26 (1962), 877–910 | Zbl

[5] Shidlovskii A. B., “K obschei teoreme ob algebraicheskoi nezavisimosti znachenii $E$-funktsii”, DAN SSSR, 171:4 (1966), 810–813

[6] Shidlovskii A. B., “Ob arifmeticheskikh svoistvakh znachenii analiticheskikh funktsii”, Trudy Matem. in-ta im. V. A. Steklova, 132, 1973, 169–202 | Zbl

[7] Shidlovskii A. B., “Ob otsenkakh mery transtsendentnosti znachenii $E$-funktsii”, Matem. zametki, 2:1 (1967), 33–44

[8] Feldman N. I., Shidlovskii A. B., “Razvitie i sovremennoe sostoyanie teorii transtsendentnykh chisel”, UMN, XXII:3 (1967), 3–81

[9] Borel E., Sur la nature arithmetique du nombre $e$, C. r. Acad. Sci., 128, Paris, 1899 | Zbl

[10] Popken J., Sur la nature aritmetique du nombre $e$, C. r. Acad. Sci. Paris, 186, 1928 | Zbl

[11] Popken J., “Zur Transzendenz von $e$”, B. 29, Math. Z., 1929, 525–541 | DOI | MR | Zbl

[12] Mahler K., “Zur Approximation der Exponentialfunktion und des Logarithmus. I”, J. reine und angew. Math., 166 (1932), 118–136

[13] Shidlovskii A. B., “Transtsendentnost znachenii $E$-funktsii”, Trudy IV Vses. matem. s'ezda, t. 2, Nauka, Leningrad, 1964, 147–158

[14] Lang S., “Transzendence measure for $E$-functions”, Mathematika, 9 (1962), 157–161 | MR | Zbl

[15] Galochkin A. I., “Otsenka mery vzaimnoi transtsendentnosti znachenii $E$-funktsii”, Matem. zametki, 3:4 (1968), 377–386 | MR | Zbl

[16] Shidlovskii A. B., “Ob otsenkakh mery transtsendentnosti znachenii $E$-funktsii”, Vestnik MGU, ser. matem., mekhanika, 1977, no. 6, 3–10

[17] Shidlovskii A. B., “On the estimates of the algebraic independence measures of the values of $E$-iunctions”, J. Austral. Math. Soc. (Ser. A), 27 (1979), 385–407 | DOI | MR

[18] Nesterenko Yu. V., “Otsenki poryadkov nulei funktsii odnogo klassa i ikh prilozheniya v teorii transtsendentnykh chisel”, Izv. AN SSSR, ser. matem., 41 (1977), 253–284 | MR | Zbl

[19] Shidlovskii A. B., “Ob otsenkakh mery transtsendentnosti znachenii $E$-funktsii”, UMN, XXII:3 (1967), 245–246

[20] Salikhov V. X., “Ob otsenkakh mer lineinoi nezavisimosti i transtsendentnosti znachenii nekotorykh $E$-funktsii v ratsionalnykh tochkakh”, Vestnik MGU, ser. matem., mekhanika, 1978, no. 3, 9–18 | MR | Zbl

[21] Makarov Yu. N., “Ob otsenkakh mery lineinoi nezavisimosti znachenii $E$-funktsii”, Vestnik MGU, ser. matem., mekhanika, 1978, no. 2, 3–12 | Zbl

[22] Väänänen K., “Estimates of the conjugate transcendence measure for the values of some $E$-functions”, Ann. Acad. Scien. Fennicae (Ser. A), Math., 536 (1973), 3–12 | MR

[23] Väänänen K., “On linear forms of the values of one class of $E$-functions”, Acta Univ. Ouluensis (Ser A), Scien. rerum natural, Math., 41:41 (1979), 403–404

[24] Shidlovskii A. B., “Ob arifmeticheskikh svoistvakh znachenii $E$-funktsii v algebraicheskikh tochkakh”, DAN SSSR, 248:1 (1979), 50–55 | MR

[25] Chirskii V. G., “Ob arifmeticheskikh svoistvakh znachenii analiticheskikh funktsii, svyazannykh algebraicheskimi uravneniyami nad polem ratsionalnykh funktsii”, Matem. zametki, 14:1 (1973), 83–94

[26] Oleinikov V. A., “O nekotorykh svoistvakh algebraicheski zavisimykh velichin”, Vestnik MGU, ser. matem., mekhanika, 1962, no. 5, 11–17 | MR | Zbl

[27] Shidlovskii A. B., “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii $E$-funktsii, svyazannykh algebraicheskim uravneniem v pole ratsionalnykh funktsii”, Vestnik MGU, matem., mekhanika, 1960, no. 25, 19–28

[28] Shidlovskii A. B., “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii nekotorykh $E$-funktsii”, Vestnik MGU, ser. matem., mekhanika, 1961, no. 5, 44–59

[29] Shidlovskii A. B., “Ob otsenke mery vzaimnoi transtsendentnosti znachenii $E$-funktsii, svyazannykh proizvolnymi algebraicheskimi uravneniyami nad $\mathbf{C}(z)$”, UMN, 35:2 (1980), 233–234 | MR

[30] Shidlovskii A. B., “Ob arifmeticheskikh svoistvakh znachenii $E$-funktsii, svyazannykh algebraicheskimi uravneniyami v pole ratsionalnykh funktsii”, DAN SSSR, 251:4 (1980), 808–812 | MR

[31] Shidlovskii A. B., “Ob arifmeticheskikh svoistvakh mnogochlenov ot znachenii $E$-funktsii, svyazannykh algebraicheskimi uravneniyami v pole ratsionalnykh funktsii”, Acta Arithmetica, 37 (1979), 403–424