Geodesic
On Newman polynomials without roots on the unit circle
Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 197-203.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this note we give a necessary and sufficient condition on the triplet of nonnegative integers $a$ for which the Newman polynomial $\sum_{j=0}^a x^j + \sum_{j=b}^c x^j$ has a root on the unit circle. From this condition we derive that for each $d \geq 3$ there is a positive integer $n>d$ such that the Newman polynomial $1+x+\dots+x^{d-2}+x^n$ of length $d$ has no roots on the unit circle.
Keywords: Newman polynomial, root of unity. 
@article{CHEB_2019_20_1_a10,
     author = {A. Dubickas},
     title = {On {Newman} polynomials without roots on the unit circle},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {197--203},
     publisher = {mathdoc},
     volume = {20},
     number = {1},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2019_20_1_a10/}
}
TY  - JOUR
AU  - A. Dubickas
TI  - On Newman polynomials without roots on the unit circle
JO  - Čebyševskij sbornik
PY  - 2019
SP  - 197
EP  - 203
VL  - 20
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2019_20_1_a10/
LA  - en
ID  - CHEB_2019_20_1_a10
ER  - 
%0 Journal Article
%A A. Dubickas
%T On Newman polynomials without roots on the unit circle
%J Čebyševskij sbornik
%D 2019
%P 197-203
%V 20
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2019_20_1_a10/
%G en
%F CHEB_2019_20_1_a10
A. Dubickas. On Newman polynomials without roots on the unit circle. Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 197-203. http://geodesic.mathdoc.fr/item/CHEB_2019_20_1_a10/

[1] D. W. Boyd, “Large Newman polynomials”, Diophantine analysis (Kensigton, 1985), London Math. Soc. Lecture Note Ser., 109, Cambridge Univ. Press, Cambridge, 1986, 159–170 | MR

[2] D. M. Campbell, H. R. P. Ferguson, R. W. Forcade, “Newman polynomials on $|z|=1$”, Indiana Univ. Math. J., 32 (1983), 517–525 | MR | Zbl

[3] A. Dubickas, “Nonreciprocal algebraic numbers of small measure”, Comment. Math. Univ. Carolin., 45 (2004), 693–697 | MR | Zbl

[4] B. Goddard, “Finite exponential series and Newman polynomials”, Proc. Amer. Math. Soc., 116 (1992), 313–320 | MR | Zbl

[5] M. Filaseta, C. Finch, C. Nicol, “On three questions concerning $0,1$-polynomials”, J. Théorie des Nombres Bordx, 118 (2006), 357–370 | MR

[6] C. Finch, L. Jones, “On the irreducibility of $\{-1,0,1\}$-quadrinomials”, Integers, 6:A16 (2006), 4 pp. | MR | Zbl

[7] Mercer I., “Newman polynomials, reducibility and roots on the unit circle”, Integers, 12:A6 (2012), 16 pp. | MR

[8] Mercer I., “Newman polynomials not vanishing on the unit circle”, Integers, 12:A67 (2012), 7 pp. | MR

[9] S. V. Konyagin, “On the number of irreducible polynomials with $0,1$ coefficients”, Acta Arith., 88 (1999), 333–350 | MR | Zbl

[10] J. J. Luo, H. J. Ruan, Y. L. Wang, “Lipschitz equivalence of Cantor sets and irreducibility of polynomials”, Mathematika, 64 (2018), 730–741 | MR | Zbl

[11] C. J. Smyth, “Some results on Newman polynomials”, Indiana Univ. Math. J., 34 (1985), 195–200 | MR | Zbl