staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Michael Filaseta; Carrie Finch; Charles Nicol
On three questions concerning ${0,1}$-polynomials
Journal de théorie des nombres de Bordeaux, 18 no. 2 (2006), p. 357-370, doi: 10.5802/jtnb.549
Article PDF | Reviews MR 2289429 | Zbl 05135395

Résumé - Abstract

We answer three reducibility (or irreducibility) questions for $0,1$-polynomials, those polynomials which have every coefficient either $0$ or $1$. The first concerns whether a naturally occurring sequence of reducible polynomials is finite. The second is whether every nonempty finite subset of an infinite set of positive integers can be the set of positive exponents of a reducible $0,1$-polynomial. The third is the analogous question for exponents of irreducible $0,1$-polynomials.

Bibliography

[1] M. Filaseta, On the factorization of polynomials with small Euclidean norm. Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997), de Gruyter, Berlin, 1999, 143–163.  MR 1689504 |  Zbl 0928.11015
[2] M. Filaseta, K. Ford, S. Konyagin, On an irreducibility theorem of A. Schinzel associated with coverings of the integers. Illinois J. Math. 44 (2000), 633–643. Article |  MR 1772434 |  Zbl 0966.11046
[3] M. Filaseta, A. Schinzel, On testing the divisibility of lacunary polynomials by cyclotomic polynomials. Math. Comp. 73 (2004), 957–965.  MR 2031418 |  Zbl 02041070
[4] W. Ljunggren, On the irreducibility of certain trinomials and quadrinomials. Math. Scand. 8 (1960), 65–70.  MR 124313 |  Zbl 0095.01305
[5] H. B. Mann, On linear relations between roots of unity. Mathematika 12 (1965), 107–117.  MR 191892 |  Zbl 0138.03102
[6] A. Schinzel, On the reducibility of polynomials and in particular of trinomials. Acta Arith. 11 (1965), 1–34. Article |  MR 180549 |  Zbl 0196.31104
[7] A. Schinzel, Selected topics on polynomials. Ann Arbor, Mich., University of Michigan Press, 1982.  MR 649775 |  Zbl 0487.12002
[8] H. Tverberg, On the irreducibility of the trinomials $x^{n} \pm x^{m} \pm 1$. Math. Scand. 8 (1960), 121–126.  MR 124314 |  Zbl 0097.00801