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

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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/}
}
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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/