On Littlewood and Newman polynomial multiples of Borwein polynomials

Drungilas, P.; Jankauskas, J.; Šiurys, J.

doi:10.1090/mcom/3258

On Littlewood and Newman polynomial multiples of Borwein polynomials
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by P. Drungilas, J. Jankauskas and J. Šiurys PDF

Math. Comp. 87 (2018), 1523-1541 Request permission

Abstract:

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 $P(X)\in \mathbb {Z}[X]$ a Borwein polynomial if all its coefficients belong to $\{-1,0,1\}$ and $P(0)\neq 0$. 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 $\mathbb {Z}[X]$ with coefficients in a finite set $\mathcal {D}\subset \mathbb {Z}$, 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.

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