Geodesic
A metric theorem on the simultaneous approximation of a zero by the values of integral polynomials
Izvestiya. Mathematics, Tome 16 (1981) no. 1, pp. 21-40 
V. I. Bernik. A metric theorem on the simultaneous approximation of a zero by the values of integral polynomials. Izvestiya. Mathematics, Tome 16 (1981) no. 1, pp. 21-40. http://geodesic.mathdoc.fr/item/IM2_1981_16_1_a1/
@article{IM2_1981_16_1_a1,
     author = {V. I. Bernik},
     title = {A metric theorem on the simultaneous approximation of a~zero by the values of integral polynomials},
     journal = {Izvestiya. Mathematics},
     pages = {21--40},
     year = {1981},
     volume = {16},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1981_16_1_a1/}
}
TY  - JOUR
AU  - V. I. Bernik
TI  - A metric theorem on the simultaneous approximation of a zero by the values of integral polynomials
JO  - Izvestiya. Mathematics
PY  - 1981
SP  - 21
EP  - 40
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/IM2_1981_16_1_a1/
LA  - en
ID  - IM2_1981_16_1_a1
ER  - 
%0 Journal Article
%A V. I. Bernik
%T A metric theorem on the simultaneous approximation of a zero by the values of integral polynomials
%J Izvestiya. Mathematics
%D 1981
%P 21-40
%V 16
%N 1
%U http://geodesic.mathdoc.fr/item/IM2_1981_16_1_a1/
%G en
%F IM2_1981_16_1_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper it is proved that the inequality $$ \prod_{i=1}^k|P(\omega_i)|^{-n+k-1-\varepsilon} $$ has only a finite number of solutions in integral polynomials $P(x)$ for almost all $\overline\omega=(\omega_1,\dots,\omega_k)$. Here $H$ is the coefficient of $P(x)$ largest in absolute value. Sprindzuk's conjecture is thereby proved. Bibliography: 7 titles.

[1] Gelfond A. O., Transtsendentnye i algebraicheskie chisla, GITTL, M., 1952

[2] Slesoraitene R., “Teorema Malera–Sprindzhuka dlya polinomov tretei stepeni ot dvukh peremennykh. II”, Litovsk. matem. sb., 10:4 (1970), 791–814 | MR

[3] Sprindzhuk V. G., Metricheskie teoremy o diofantovykh priblizheniyakh i priblizheniyakh algebraicheskimi elementami ogranichennoi stepeni, dissertatsiya, LGU im. A. A. Zhdanova, 1963

[4] Sprindzhuk V. G., “Dokazatelstvo gipotezy Malera o mere mnozhestva $S$-chisel”, Izv. AN SSSR. Ser. matem., 29 (1965), 379–436

[5] Sprindzhuk V. G., Problema Malera v metricheskoi teorii chisel, Nauka i tekhnika, Minsk, 1967

[6] Baker A., “On Theorem of Sprindzuk”, Proc. Royal Soc.(A), 292:1428 (1966), 92–104 | DOI | Zbl

[7] Wirsing E., “Approximation mit algebraischen Zahlen beschrankten Grades”, J. reine und angewandte Math., 206:1–2 (1961), 67–77 | MR | Zbl