On three questions concerning 0,1-polynomials
Michael Filaseta; Carrie Finch; Charles Nicol
Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 2, p. 357-370

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.

Nous répondons à trois questions concernant la réductibilité (ou irréductibilité) de 0,1-polynômes, polynômes qui n’ont pour seuls coefficients que 0 ou 1. La première question est de déterminer si une suite de polynômes qui se présente naturellement est finie. Deuxièmement, nous discutons si tout sous-ensemble fini d’un ensemble infini de nombres entiers positifs peut être l’ensemble des exposants d’un 0,1-polynôme réductible. La troisième question est similaire, mais pour l’ensemble des exposants d’un polynôme irréductible.

Received : 2004-11-16
Published online : 2008-12-03
DOI : https://doi.org/10.5802/jtnb.549
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     author = {Michael Filaseta and Carrie Finch and Charles Nicol},
     title = {On three questions concerning ${0,1}$-polynomials},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {2},
     year = {2006},
     pages = {357-370},
     doi = {10.5802/jtnb.549},
     mrnumber = {2289429},
     zbl = {05135395},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2006__18_2_357_0}
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Filaseta, Michael; Finch, Carrie; Nicol, Charles. On three questions concerning ${0,1}$-polynomials. Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 2, pp. 357-370. doi : 10.5802/jtnb.549. jtnb.centre-mersenne.org/item/JTNB_2006__18_2_357_0/

[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. | 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. | 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 ±x m ±1. Math. Scand. 8 (1960), 121–126. | MR 124314 | Zbl 0097.00801