Uspekhi Mat. Nauk 45:1 (1990), 163-165

Russian Math. Surveys 45:1 (1990), 191-194

Vyacheslav Alekseevich Oleinikov(obituary)
On the night of 4 — 5 March 1989 Vyacheslav Alekseevich Oleinikov,
professor of the theory of functions and functional analysis in the faculty ofmechanics and mathematics of Moscow State University, died tragically in afire at his house outside Moscow.
Oleinikov was born on 10 March 1939 in Moscow. In 1956 he enrolled
in the faculty of mechanics and mathematics of Moscow State University.
After graduating (in 1961) he worked for two years in the Computer
Laboratory at Moscow State University (and later on he kept up his interestin applied problems: he published several papers on work connected witheconomic treaties). In 1963 he became a research student in the faculty ofmechanics and mathematics of Moscow State University in the departmentof number theory (of which A.O. Gel'fond was then the director) and his
supervisor was A.B. Shidlovskii.
Oleinikov's main research interests were problems of transcendental
numbers, which are closely linked to algebraic and classical transcendental
functions and differential equations. In 1929 C.L. Siegel propounded amethod (based on the ideas of Hermite and Lindemann) which made it
possible to study the arithmetical nature of the values at algebraic points ofquite a wide class of entire analytic functions (Siegel's ^-functions), which
satisfy linear differential equations with polynomial coefficients. In 1955
Shidlovskii developed Siegel's ideas and completely reduced the arithmeticalproblem of the transcendental nature and algebraic independence of thevalues of such functions at algebraic points to the function-theoretic
independence of these same functions (and their derivatives) over the field ofrational functions. However in real situations the application of Shidlovskii's
theorem ran into significant difficulties; in particular, it was not possible toinvestigate fully the question of the independence of y(oc), /(a), /'(a) for theclassical i?-function

Σ/ % y»nιη! (λ -h 1) . . . (λ Η- ») (μ -f- 1) . . . (μ -Η »)\~Τ)

satisfying the third-order equation'λ (λ + μ Η 1)
(3λ + 1) (3μ +1)</" -l·:ν" \7ιy —y = °'192

V.A. Oleinikov

with rationals λ, μ Φ - 1 , -2, ... . Oleinikov considered the general conceptof the differential reducibility of an algebraic differential equation and gave
criteria for reducibility, reducing the question to one of the solution in
algebraic functions of certain partial differential equations, explicitly generated
by the initial equation. In the end, by applying this to the third-orderequation he was able to prove that for algebraic α Φ 0 the numbersy(<x), / ( a ) , /'(a) &re algebraically dependent if and only if λ + 1/3 andμ + 1/3 are simultaneously integers. This approach proved to be applicable
to some other classical hypergeometric functions.
We must mention that Oleinikov's research on differential irreducibility,
which he continued to the end of his life (regrettably his last results were only
published in summary form), is clearly of interest in the context of numbertheory.
Oleinikov's other research was directly associated with the study of algebraic
functions of several variables. He introduced the concept of an algebraic
function with separated singularities and he gave explicit representations ofsuch functions. Results on rational algebraic superpositions are close to thiswork.
Oleinikov was quite sceptical about fashion in mathematics; he knew andloved the classical mathematics of the 19th century (his main papers usually
begin with a reference to Hermite or Frobenius). That may explain why hismathematical contacts were rather narrow.
From the time he received his degree of Doctor of Philosophy until the endof his life Oleinikov worked in the department of the theory of functions andfunctional analysis in the faculty of mechanics and mathematics of MoscowState University. Although he took part in almost all forms of teaching—giving lecture courses (including very original courses on the theory of specialfunctions), supervising research students and undergraduates, holding special
seminars, he enjoyed above all else giving lectures to second-year students on
the theory of analytic functions. Here he succeeded in presenting some originalideas, which required accurate thinking through and precise execution. For
example, all the basic material for the first stage was covered in the firstsemester, and in the second one he introduced more general questions; thisapproach, which of course requires extra effort, usually meant that theinteresting and exciting sections had to be considered hurriedly at the end of
the academic year. Over his 25 years of teaching many hundreds of graduatesfrom the faculty of mechanics and mathematics remember him with gratitude,even though he was strict and demanding. He worked with his pupils, heloved elementary mathematics, and he wrote interesting articles for "Kvant".
Oleinikov was an unusual man; he had a wide range of interests, he was aprofound scholar and a connoisseur of works on art, history, and philosophy.
He loved and understood music, he played the piano quite well even though
he had never learnt it—he simply had an instrument and some scores; hepreferred Chopin, Liszt, and Beethoven. In one of the summer months heV.A. Oleinikov

193

used to go with his family and friends on a canoeing expedition. He did not
like hunting, he was not an enthusiastic fisherman, but after years of travelling
along the rivers and lakes of our country he had collected and classified a finecollection of butterflies and beetles.
For Oleinikov nothing in life or in work or in public life was too trivial.
A true and reliable friend, amusing and witty, he had a deep and subtle
understanding of people, he was able quickly and accurately to distinguishbetween the apparent and the real, he was intolerant of any compromise thatwent against his conscience. It was natural that in a situation of moral
uncertainty it was not easy for him, but nor was it always easy for people
around him to deal with him. We must acknowledge that astonishingphenomenon of Vyacheslav Alekseevich Oleinikov: in critical situations bypersistently and stubbornly championing his uncomfortable truth he very often
won a victory, even though it was at the price of his peace of mind.
Moscow University has lost one of its dedicated colleagues, an outstanding
teacher and gifted scientist, his wife and children have lost a true and loving
husband and father, and all of us who knew him have lost a vital part of oureveryday life. His shining memory lives on.
V.I. Arnol'd, E.A. Gorin, V.Ya. Lin,Yu.V. Nesterenko, A.B. Shidlovskii,M.A. Shubin, V.M. Tikhomirov,P.L. Ul'yanov, A.G. Vitushkin
List of V.A. Oleinikov's published papers[1] Some properties of algebraically dependent quantities, Vestnik Moskov. Univ. Ser. 1
1962, no. 5, 11-17. MR 26 # 1334.
[2] The transcendence and algebraic independence of the values of certain ^-functions.
Vestnik Moskov. Univ. Ser. 1 1962, no. 6, 34-38. MR 26 # 1285.
[3] The transcendence and algebraic independence of values of ^-functions representing asolution of a third-order linear differential equation, Dokl. Akad. Nauk SSSR 166(1966), 540-543.
= Soviet Math. Dokl. 7 (1966), 118-121.
[4] Algebraic independence of values of £-functions satisfying linear non-homogeneous
differential equations of the third order, Dokl. Akad. Nauk SSSR 169 (1966), 32-34.
MR 33 # 7303.
= Soviet Math. Dokl. 7 (1966), 869-871.
[5] The transcendence and algebraic independence of the values of certain entire
functions, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 63-92. MR 36 # 5085.
= Math. USSR-Izv. 2 (1968), 61-87.
[6] Algebraic functions with the simplest arrangement of singularities, Vestnik Moskov.
Univ. Ser. 1 1968, no. 6, 9-13 (with A.N. Zhdanov). MR 40 # 7473.
[7] The algebraic independence of the values of ^-functions, Mat. Sb. 78 (1969),
301-306. MR 38 # 4418.
= Math. USSR-Sb. 7 (1989), 293-298.

194

V.A. Oleinikov[8] Certain properties of differential polynomials, Vestnik Moskov. Univ. Ser. 1 1969
no. 3, 50-58. MR 40 # 7233.
= Moscow Univ. Math. Bull 24:3/4 (1969), 41-47.
[9] The differential irreducibility of a linear inhomogeneous equation, Dokl. Akad. NaukSSSR 194 (1970), 1017-1020. MR 43 # 194.
= Soviet Math. Dokl. 11 (1970), 1337-1341.
[10] The differential irreducibility of linear equations, Vestnik Moskov Univ. Ser. 1 1970,
no. 1, 44-51. MR 43 # 7419.
= Moscow Univ. Math. Bull 26:1/2 (1971), 35-41.
[11] A sufficient criterion for differential irreducibility of a linear equation, Vestnik
Moskov. Univ. Ser. 1 1971, no. 3, 45-52. MR 43 # 7420.
= Moscow Univ. Math. Bull. 26:3/4 (1971), 35-40.
[12] Algebraic functions with separated singularities, Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 1294-1315. MR 45 # 5136.
= Math. USSR-Izv. 5 (1971), 1303-1324.
[13] Rational algebraic superpositions, Dokl. Akad. Nauk SSSR 200 (1971), 1283-1286.
MR 47 # 191.
= Soviet Math. Dokl. 12 (1971), 1577-1581.
[14] Conditions for mutual transcendence of hypergeometric series, Vestnik Moskov. Univ.
Ser. 1 1986, no. 1, 20-25. MR 87g: 11087.
= Moscow Univ. Math. Bull. 41:1 (1986), 26-32.
Translated by A. Lofthouse