Alexander Ossipowitsch Gelfond

Alexander Gelfond ( Russian Александр Осипович Гельфонд * 11 ^{. Jul} / 24. October  1906 ^greg. In Saint Petersburg , † 7. November 1968 in Moscow ) was a Russian mathematician.

Life

Gelfond was born on October 24, 1906, the son of a doctor. He studied from 1924 to 1927 at Moscow University and continued his postgraduate training with Alexander Chintschin (1894-1959) and Vyacheslav Stepanov (1889-1950). After brief teaching at the Moscow Technical University, he became a professor of analysis in 1931 , and later for number theory at Moscow University. He held this position until his death, from 1933 onwards he worked at the Moscow Steklov Institute for Mathematics. In 1935 he earned a doctorate in mathematics and physics. Vera Nikolaevna Maslennikova was a student of Gelfond.

Gelfond achieved excellent results especially in the field of number theory. He further developed the traditions of Russian-Soviet mathematics in this field and became a co-founder of a successful Soviet number theory school. As early as 1929 he discovered deep-seated connections between the growth and other properties of entire analytical functions and the arithmetic of their values ​​and thus solved Hilbert's seventh problem for a special case, including the proof of the transcendence of . After improving his method ( Gelfonds second method ), u. a. considering linear forms of the exponential functions , he succeeded in 1934 with the proof that for an algebraic number a ≠ 0 or 1, and an algebraic, irrational number b, the number is transcendent ( theorem of Gelfond-Schneider ). Independently of this, Hilbert's seventh problem was solved only a little later by Theodor Schneider (1911–1988). An obvious generalization of the theorem could not be proven until 1966 by Alan Baker . Gelfond's results and the application of his methods led to significant advances in the theory of transcendent numbers. He constructed new classes of transcendent numbers, solved questions regarding the mutual algebraic independence of numbers and successfully extended the method to p-adic functions. In 1952 he summarized many results in a monograph "Transzendentnyje i algebraitscheskije čisla" (Трансцендентные и алгебраические числа). ${\displaystyle e^{\pi }}$ ${\displaystyle a^{b}}$

A conjecture by Gelfond, which has not yet been solved, extends Gelfond-Schneider's theorem to include systems of numbers ( ) and assumes that these are algebraically independent of one another over the rational numbers, if the algebraic numbers are linearly independent over the rational numbers . The best result was achieved in 1989 by G. Diaz, who showed that the degree of transcendence of the system of numbers (with an irrational one ) is at least . Gelfond proved itself a special case (n = 2 , a cubic irrational number). ${\displaystyle {\alpha }^{{\beta }_{1}},...,{\alpha }^{{\beta }_{n}}}$ ${\displaystyle \alpha \neq 0,1}$ ${\displaystyle 1,{\beta }_{1},...,{\beta }_{n}}$ ${\displaystyle {\alpha }^{{\beta }_{i}}}$ ${\displaystyle {\beta }_{i}={{\beta }_{1}}^{i}}$ ${\displaystyle {\beta }_{1}}$ ${\displaystyle {\frac {n+2}{2}}}$ ${\displaystyle {\beta }_{2}={{\beta }_{1}}^{2}}$ ${\displaystyle {\beta }_{1}}$

Gelfonds' research on the interpolation and approximation of functions of a complex variable was partly closely linked to the number theoretical studies . He examined in detail the convergence of interpolation methods depending on the set of given points and the properties of the function to be approximated, as well as the unambiguous determination of the constructed function. He also compiled these results in 1952 in the monograph, which appeared in 1958 as a difference calculation in German translation as Volume 41 of the university books for mathematics . Further topics were the completeness of function systems and the asymptotic behavior of the eigenvalues ​​of certain integral equations.

In addition, the mathematics history of Gelfonds found interest, he promoted it above all with studies on the number theoretical work of Euler .

literature

• BV Levin, NI Feldman, AB Šidlovski: Alexander O. Gelfond ( PDF file, 1.1 MB), Acta Arithmetica 17, 1971, pp. 314–336 (English; obituary)

Web links

• Literature by and about Alexander Ossipowitsch Gelfond in the catalog of the German National Library
• John J. O'Connor, Edmund F. Robertson :  Alexander Ossipowitsch Gelfond. In: MacTutor History of Mathematics archive .

Individual evidence

1. ↑ Feldman, Algebraic and transcentdal numbers, Quantum, July / August 2000 <