Geodesic
On the number of integral polynomialswith bounds placed on the derivative at $p$-adic and real roots
Trudy Instituta matematiki, Tome 25 (2017) no. 2, pp. 6-10 
M. L. Bezrukov. On the number of integral polynomialswith bounds placed on the derivative at $p$-adic and real roots. Trudy Instituta matematiki, Tome 25 (2017) no. 2, pp. 6-10. http://geodesic.mathdoc.fr/item/TIMB_2017_25_2_a0/
@article{TIMB_2017_25_2_a0,
     author = {M. L. Bezrukov},
     title = {On the number of integral polynomialswith bounds placed on the derivative at $p$-adic and real roots},
     journal = {Trudy Instituta matematiki},
     pages = {6--10},
     year = {2017},
     volume = {25},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2017_25_2_a0/}
}
TY  - JOUR
AU  - M. L. Bezrukov
TI  - On the number of integral polynomialswith bounds placed on the derivative at $p$-adic and real roots
JO  - Trudy Instituta matematiki
PY  - 2017
SP  - 6
EP  - 10
VL  - 25
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMB_2017_25_2_a0/
LA  - ru
ID  - TIMB_2017_25_2_a0
ER  - 
%0 Journal Article
%A M. L. Bezrukov
%T On the number of integral polynomialswith bounds placed on the derivative at $p$-adic and real roots
%J Trudy Instituta matematiki
%D 2017
%P 6-10
%V 25
%N 2
%U http://geodesic.mathdoc.fr/item/TIMB_2017_25_2_a0/
%G ru
%F TIMB_2017_25_2_a0

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

Consider a class of polynomials defined by a fixed degree and a fixed height. Introducing an additional constraint on the value of the $p$-adic norm of the derivative at a $p$-adic root, we find an upper bound on the number of such polynomials. A similar bound has been proved in the case where the derivative is bounded at a real and a $p$-adic root.

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

[2] Bernik V. I., Vasilev D. V., Kudin A. C., “O chisle tselochislennykh mnogochlenov zadannoi stepeni i ogranichennoi vysoty s maloi proizvodnoi v korne mnogochlena”, Trudy Instituta matematiki, 22:2 (2014), 3–8 | MR

[3] Koblits N., $p$-Adicheskie chisla, $p$-adicheskii analiz, Mir, M., 1982 | MR

[4] Bernik V. I., “Sovmestnye priblizheniya nulya znacheniyami tselochislennykh mnogochlenov”, Doklad AN SSSR, 44:1 (1980), 24–45 | Zbl

[5] Baker R. C., “Sprindzuk's theorem and Hausdorff dimension”, Mathematika, 23:2 (1976), 184–197 | DOI | MR | Zbl

[6] Bernik V. I., “Primenenie razmernosti Khausdorfa v teorii diofantovykh priblizhenii”, Acta Arithmetica, 42:3 (1983), 219–253 | DOI | MR | Zbl