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- W2012963516 abstract "Recall that a polynomial $P(x) in mathbb {Z}[x]$ with coefficients $0, 1$ and constant term $1$ is called a Newman polynomial, whereas a polynomial with coefficients $-1, 1$ is called a Littlewood polynomial. Is there an algebraic number $alpha$ which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no Littlewood polynomial? In this paper, for each Newman polynomial $P$ of degree at most $8,$ we find a Littlewood polynomial divisible by $P$. Moreover, it is shown that every trinomial $1+ux^a+vx^b,$ where $a<b$ are positive integers and $u, v in {-1,1},$ so, in particular, every Newman trinomial $1+x^a+x^b,$ divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., $x^9+x^6+x^2+x+1.$ This example settles the problem 006:07 posed by the first named author at the 2006 West Coast Number Theory conference. It also shows that the sets of roots of Newman polynomials $V_{mathcal {N}}$, Littlewood polynomials $V_{mathcal {L}}$ and ${-1,0,1}$ polynomials $V$ are distinct in the sense that between them there are only trivial relations $V_{mathcal {N}}subset V$ and $V_{mathcal {L}}subset V.$ Moreover, $V ne V_{mathcal {L}} cup V_{mathcal {N}}.$ The proofs of several main results (after some preparation) are computational." @default.
- W2012963516 created "2016-06-24" @default.
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- W2012963516 date "2009-01-01" @default.
- W2012963516 modified "2023-10-18" @default.
- W2012963516 title "On Newman polynomials which divide no Littlewood polynomial" @default.
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- W2012963516 doi "https://doi.org/10.1090/s0025-5718-08-02138-8" @default.
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