Geodesic
On the Distribution of Irreducible Trinomials
Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 748-756

Voir la notice de l'article provenant de la source Cambridge University Press

We obtain new results about the number of trinomials ${{t}^{n}}\,+\,at\,+\,b$ with integer coefficients in a box $(a,\,b)\,\in \,[C,\,C\,+\,A]\,\times \,[D,\,D\,+\,B]$ that are irreducible modulo a prime $p$ . As a by-product we show that for any $p$ there are irreducible polynomials of height at most ${{p}^{1/2+o(1)}}$ , improving on the previous estimate of ${{p}^{2/3+o(1)}}$ obtained by the author in 1989.
DOI : 10.4153/CMB-2011-053-0
Mots-clés : 11L40, 11T06, irreducible trinomials, character sums 
Shparlinski, Igor E. On the Distribution of Irreducible Trinomials. Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 748-756. doi: 10.4153/CMB-2011-053-0
@article{10_4153_CMB_2011_053_0,
     author = {Shparlinski, Igor E.},
     title = {On the {Distribution} of {Irreducible} {Trinomials}},
     journal = {Canadian mathematical bulletin},
     pages = {748--756},
     year = {2011},
     volume = {54},
     number = {4},
     doi = {10.4153/CMB-2011-053-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-053-0/}
}
TY  - JOUR
AU  - Shparlinski, Igor E.
TI  - On the Distribution of Irreducible Trinomials
JO  - Canadian mathematical bulletin
PY  - 2011
SP  - 748
EP  - 756
VL  - 54
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-053-0/
DO  - 10.4153/CMB-2011-053-0
ID  - 10_4153_CMB_2011_053_0
ER  - 
%0 Journal Article
%A Shparlinski, Igor E.
%T On the Distribution of Irreducible Trinomials
%J Canadian mathematical bulletin
%D 2011
%P 748-756
%V 54
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-053-0/
%R 10.4153/CMB-2011-053-0
%F 10_4153_CMB_2011_053_0

[1] [1] Adleman, L. M. and Lenstra, H. W., Finding irreducible polynomials over finite fields. In: Proc. 18th ACM Symp. Theory Comput. (Berkeley, 1986), ACM, New York, 1986, 350–355. Google Scholar

[2] [2] Ayyad, A., Cochrane, T. and Zheng, Z., The congruence x x ≡ x x (mod p), the equation x x = x x and the mean value of character sums. J. Number Theory 59(1996), 398–413. doi:10.1006/jnth.1996.0105 Google Scholar

[3] [3] Banks, W. D. and Shparlinski, I. E., Sato–Tate, cyclicity, and divisibility statistics on average for elliptic curves of small height.. Israel J. Math. 173(2009), 253–277. doi:10.1007/s11856-009-0091-0 Google Scholar

[4] [4] Cohen, S. D., The distribution of polynomials over finite fields.. Acta Arith. 17(1970), 255–271. Google Scholar

[5] [5] Cohen, S. D., Uniform distribution of polynomials over finite fields.. J. London Math. Soc. 6(1972), 93–102. doi:10.1112/jlms/s2-6.1.93 Google Scholar

[6] [6] Friedlander, J. B. and Iwaniec, H., The divisor problem for arithmetic progressions.. Acta Arith. 45(1985), 273–277. Google Scholar

[7] [7] Iwaniec, H. and Kowalski, E., Analytic number theory. Amer. Math. Soc., Providence, RI, 2004. Google Scholar

[8] [8] Shparlinski, I. E., Distribution of primitive and irreducible polynomials modulo a prime.. (Russian) Diskret. Mat. 1(1989), 117–124; translation in Discrete Math. Appl. (1991), 59–67. Google Scholar

[9] [9] Shparlinski, I. E., On irreducible polynomials of small height in finite fields. Appl. Algebra Engrg. Comm. Comput. 4(1996), no. 6, 427–431. doi:10.1007/s002000050043 Google Scholar

[10] [10] Shparlinski, I. E., Finite fields: Theory and computation. Kluwer Acad. Publ., Dordrecht, 1999. Google Scholar

Cité par Sources :