Geodesic
On polynomials of prescribed height in finite fields
Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 247-255 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper deals with the set $\mathfrak M(B)$ of monic polynomials of degree $n$ with integral coefficients belonging to a given $n$-dimensional cube $B$ with side $h$. An asymptotic formula is obtained for the number of polynomials in $\mathfrak M(B)$ having a specific type of decomposition into irreducible factors modulo some prime $p$, and an asymptotic formula for the number of primitive polynomials modulo $p$ in $\mathfrak M(B)$, which translates when $n=1$ into known results of I. M. Vinogradov on the distribution of primitive roots. These asymptotic formulas are nontrivial when $h\geqslant p^{n/(n+1)+\varepsilon}$ for any $\varepsilon>0$. Moreover, an asymptotic formula is obtained for the average value of the number of divisors modulo $p$ of polynomials in $\mathfrak M(B)$, a result that is nontrivial when $h\geqslant\max(p^{1-2/n}\ln p,p^{1/2}\ln p)$. Bibliography: 11 titles. 
@article{SM_1989_63_1_a16,
     author = {I. E. Shparlinski},
     title = {On polynomials of prescribed height in finite fields},
     journal = {Sbornik. Mathematics},
     pages = {247--255},
     year = {1989},
     volume = {63},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1989_63_1_a16/}
}
TY  - JOUR
AU  - I. E. Shparlinski
TI  - On polynomials of prescribed height in finite fields
JO  - Sbornik. Mathematics
PY  - 1989
SP  - 247
EP  - 255
VL  - 63
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1989_63_1_a16/
LA  - en
ID  - SM_1989_63_1_a16
ER  - 
%0 Journal Article
%A I. E. Shparlinski
%T On polynomials of prescribed height in finite fields
%J Sbornik. Mathematics
%D 1989
%P 247-255
%V 63
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1989_63_1_a16/
%G en
%F SM_1989_63_1_a16
I. E. Shparlinski. On polynomials of prescribed height in finite fields. Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 247-255. http://geodesic.mathdoc.fr/item/SM_1989_63_1_a16/

[1] Cohen S. D., “The distribution of polinomial over finite fields”, Acta Arithm., 17 (1970), 255–271 | MR | Zbl

[2] Cohen S. D., “Uniform distribution of polinomial over finite fields”, J. London Math. Soc., 6 (1972), 93–102 | DOI | MR | Zbl

[3] Stepanov S. A., “O chisle neprivodimykh nad konechnym polem mnogochlenov zadannogo vida”, Matem. zametki, 41:3 (1987), 289–295 | MR | Zbl

[4] Vinogradov I. M., Osnovy teorii chisel, Nauka, M., 1981 | MR

[5] Shparlinskii I. E., “O koeffitsientakh primitivnykh mnogochlenov”, Matem. zametki, 38:6 (1985), 810–815 | MR

[6] Kurosh A. G., Kurs vysshei algebry, Nauka, M., 1975 | Zbl

[7] Weil A., “On some exponential sums”, Proc. Nat. Acad. Sci. USA, 34 (1948), 204–207 | DOI | MR | Zbl

[8] Cohen S. D., “Some aritmetical functions in finite fields”, Glasgow Math. J., 11 (1970), 21–36 | DOI | MR | Zbl

[9] Gekke E., Lektsii po teorii algebraicheskikh chisel, Gostekhizdat, M., 1940

[10] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1985 | MR | Zbl

[11] Chela R., “Reducible polynomials”, J. London Math. Soc., 38 (1963), 183–188 | DOI | MR | Zbl