Speciality:
01.01.06 (Mathematical logic, algebra, and number theory)
Birth date:
13.08.1915
Keywords: diophantine approximations; transcendental numbers.
Subject:
The problem to prove the algebraic independence of values at algebraic points of a set of Siegel's E-functions, which constitutes a solution of system of linear differential equations with coefficients from $C(z)$ and which consists of functions, algebraically independent over $C(z)$, is solved. In the case of the algebraic dependence over $C(z)$ of the considered E-functions a general theorem on the transcendence degree of the set of values of these functions at algebraic points is established. Theorems about the algebraic independence of subsets of the set of these values are proved. Lower bounds for the moduli of linear forms and polynomials with integer coefficients in the values of the considered E-functions are found. The general theorems are applied to a series of concrete subsets of E-functions.
Main publications:
Shidlovskii A. B. O kriterii algebraicheskoi nezavisimosti znachenii odnogo klassa tselykh funktsii // Izv. AN SSSR. Ser. matem. 1959. T. 23, # 1, s. 35–66.
Shidlovskii A. B. O transtsendentnosti i algebraicheskoi nezavisimosti znachenii E-funktsii, svyazannykh lyubym chislom algebraicheskikh uravnenii nad polem ratsionalnykh funktsii // Izv. AN SSSR. Ser. matem. 1962. T. 26, # 6, s. 877–910.
Shidlovskii A. B. K obschei teoreme ob algebraicheskoi nezavisimosti znachenii E-funktsii // DAN SSSR. 1966. T. 171, # 4, s. 810–813.
Shidlovskii A. B. Ob otsenkakh mnogochlenov ot znachenii E-funktsii // Matem. sb. 1981. T. 115 (157), # 1 (5), s. 3–39.
