Computational experiences on the distances of polynomials to irreducible polynomials
Authors:
A. Bérczes and L. Hajdu
Journal:
Math. Comp. 66 (1997), 391-398
MSC (1991):
Primary 11C08, 11R09; Secondary 11T06, 11Y99
DOI:
https://doi.org/10.1090/S0025-5718-97-00801-6
MathSciNet review:
1377660
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we deal with a problem of Turán concerning the ‘distance’ of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial $P\in$ ${\mathbb {Z}}[x]$ of degree $\leq 22$ there exists a monic polynomial $Q\in {\mathbb {Z}}[x]$ with deg($Q$) = deg($P$) such that $Q$ is irreducible over $\mathbb {Q}$ and the ‘distance’ of $P$ and $Q$ is $\leq 4$.
- Stephen D. Cohen, The distribution of polynomials over finite fields, Acta Arith. 17 (1970), 255–271. MR 277501, DOI https://doi.org/10.4064/aa-17-3-255-271
- Stephen D. Cohen, The distribution of polynomials over finite fields. II, Acta Arith. 20 (1972), 53–62. MR 291135, DOI https://doi.org/10.4064/aa-20-1-53-62
- Stephen D. Cohen, Uniform distribution of polynomials over finite fields, J. London Math. Soc. (2) 6 (1972), 93–102. MR 309906, DOI https://doi.org/10.1112/jlms/s2-6.1.93
- K. Győry, On the irreducibility of neighbouring polynomials, Acta Arith. 67 (1994), no. 3, 283–294. MR 1292740, DOI https://doi.org/10.4064/aa-67-3-283-294
- David R. Hayes, The distribution of irreducibles in ${\rm GF}[q,\,x]$, Trans. Amer. Math. Soc. 117 (1965), 101–127. MR 169838, DOI https://doi.org/10.1090/S0002-9947-1965-0169838-6
- Rudolf Lidl and Harald Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, 1986. MR 860948
- Ilene H. Morgan and Gary L. Mullen, Primitive normal polynomials over finite fields, Math. Comp. 63 (1994), no. 208, 759–765, S19–S23. MR 1257578, DOI https://doi.org/10.1090/S0025-5718-1994-1257578-3
- Svein Mossige, Table of irreducible polynomials over ${\rm GF}[2]$ of degrees $10$ through $20$, Math. Comp. 26 (1972), 1007–1009. MR 313227, DOI https://doi.org/10.1090/S0025-5718-1972-0313227-5
- W. Wesley Peterson and E. J. Weldon Jr., Error-correcting codes, 2nd ed., The M.I.T. Press, Cambridge, Mass.-London, 1972. MR 0347444
- A. Schinzel, Reducibility of polynomials and covering systems of congruences, Acta Arith. 13 (1967/68), 91–101. MR 219515, DOI https://doi.org/10.4064/aa-13-1-91-101
- A. Schinzel, Reducibility of lacunary polynomials. II, Acta Arith. 16 (1969/70), 371–392. MR 265323, DOI https://doi.org/10.4064/aa-16-4-371-392
Retrieve articles in Mathematics of Computation with MSC (1991): 11C08, 11R09, 11T06, 11Y99
Retrieve articles in all journals with MSC (1991): 11C08, 11R09, 11T06, 11Y99
Additional Information
A. Bérczes
Affiliation:
Department of Mathematics and Informatics, Kossuth Lajos University, 4010 Debrecen, Pf. 12, Hungary
Email:
berczes@dragon.klte.hu
L. Hajdu
Affiliation:
Department of Mathematics and Informatics, Kossuth Lajos University, 4010 Debrecen, Pf. 12, Hungary
MR Author ID:
339279
Email:
hajdul@math.klte.hu
Received by editor(s):
July 19, 1995
Received by editor(s) in revised form:
February 2, 1996
Additional Notes:
Research of the second author was supported in part by Grants 014245 and T 016 975 from the Hungarian National Foundation for Scientific Research, by the Universitas Foundation of Kereskedelmi Bank RT and by Foundation for Hungarian Higher Education and Research.
Article copyright:
© Copyright 1997
American Mathematical Society