On Littlewood and Newman polynomial multiples of Borwein polynomials
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In: Mathematics of computation, Vol. 87.2018, No. 311, 2018, p. 1523-1541.
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TY - JOUR
T1 - On Littlewood and Newman polynomial multiples of Borwein polynomials
AU - Drungilas, Paulius
AU - Jankauskas, Jonas
AU - Šiurys, Jonas
PY - 2018
Y1 - 2018
N2 - A Newman polynomial has all the coefficients in {0,1} and constant term 1, whereas a Littlewood polynomial has all coefficients in {-1,1}. We call P(X) in Z[X] a Borwein polynomial if all its coefficients belong to {-1,0,1} and P(0) not equal to 0. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle |z|=1 has a non-zero multiple in Z[X] with coefficients in a finite set D subset Z, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.
AB - A Newman polynomial has all the coefficients in {0,1} and constant term 1, whereas a Littlewood polynomial has all coefficients in {-1,1}. We call P(X) in Z[X] a Borwein polynomial if all its coefficients belong to {-1,0,1} and P(0) not equal to 0. By exploiting an algorithm which decides whether a given monic integer polynomial with no roots on the unit circle |z|=1 has a non-zero multiple in Z[X] with coefficients in a finite set D subset Z, for every Borwein polynomial of degree at most 9 we determine whether it divides any Littlewood or Newman polynomial. In particular, we show that every Borwein polynomial of degree at most 8 which divides some Newman polynomial divides some Littlewood polynomial as well. In addition to this, for every Newman polynomial of degree at most 11, we check whether it has a Littlewood multiple, extending the previous results of Borwein, Hare, Mossinghoff, Dubickas and Jankauskas.
KW - Borwein polynomials
KW - Littlewood polynomials
KW - Newman
KW - Pisot numbers
KW - Salem Numbers
KW - Mahler's measure
KW - Polynomials of small height
U2 - 10.1090/mcom/3258
DO - 10.1090/mcom/3258
M3 - Article
VL - 87.2018
SP - 1523
EP - 1541
JO - Mathematics of computation
JF - Mathematics of computation
SN - 0025-5718
IS - 311
ER -