On Newman polynomials which divide no Littlewood polynomial

Dubickas, Artūras; Jankauskas, Jonas

doi:10.1090/S0025-5718-08-02138-8

On Newman polynomials which divide no Littlewood polynomial
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by Artūras Dubickas and Jonas Jankauskas PDF

Math. Comp. 78 (2009), 327-344 Request permission

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.

References

Francesco Amoroso, Polynomials with prescribed vanishing at roots of unity, Boll. Un. Mat. Ital. B (7) 9 (1995), no. 4, 1021–1042 (English, with Italian summary). MR 1369388
Frank Beaucoup, Peter Borwein, David W. Boyd, and Christopher Pinner, Multiple roots of $[-1,1]$ power series, J. London Math. Soc. (2) 57 (1998), no. 1, 135–147. MR 1624809, DOI 10.1112/S0024610798005857
Franck Beaucoup, Peter Borwein, David W. Boyd, and Christopher Pinner, Power series with restricted coefficients and a root on a given ray, Math. Comp. 67 (1998), no. 222, 715–736. MR 1468939, DOI 10.1090/S0025-5718-98-00960-0
D. A. Bini and G. Fiorentino, Numerical computation of polynomial roots v. 2.0, FRISCO report (1998) (available online at http://www.dm.unipi.it/cluster-pages/ mpsolve/ index. htm).
P. Borwein, E. Dobrowolski and M. J. Mossinghoff, Lehmer’s problem for polynomials with odd coefficients, Ann. of Math. (2), 166 (2007), 347–366.
Peter Borwein, Kevin G. Hare, and Michael J. Mossinghoff, The Mahler measure of polynomials with odd coefficients, Bull. London Math. Soc. 36 (2004), no. 3, 332–338. MR 2038720, DOI 10.1112/S002460930300287X
Peter Borwein and Michael J. Mossinghoff, Polynomials with height 1 and prescribed vanishing at 1, Experiment. Math. 9 (2000), no. 3, 425–433. MR 1795875
Peter Borwein and Christopher Pinner, Polynomials with $\{0,+1,-1\}$ coefficients and a root close to a given point, Canad. J. Math. 49 (1997), no. 5, 887–915. MR 1604114, DOI 10.4153/CJM-1997-047-3
P. Drungilas and A. Dubickas, Roots of polynomials of bounded height, Rocky Mt. J. Math. (to appear).
Artūras Dubickas and Michael J. Mossinghoff, Auxiliary polynomials for some problems regarding Mahler’s measure, Acta Arith. 119 (2005), no. 1, 65–79. MR 2163518, DOI 10.4064/aa119-1-5
Michael Filaseta, On the factorization of polynomials with small Euclidean norm, Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 143–163. MR 1689504
Michael Filaseta and Manton Matthews Jr., On the irreducibility of $0,1$-polynomials of the form $f(x)x^n+g(x)$, Colloq. Math. 99 (2004), no. 1, 1–5. MR 2084532, DOI 10.4064/cm99-1-1
W. Hofschuster and W. Krämer, C–XSC 2.0: A C++ Class Library for Extended Scientific Computing, Numerical Software with Result Verification, Lecture Notes in Computer Science, 2991, Springer-Verlag, Heidelberg, (2004) 15–35 (available online at http://www.math.uni-wuppertal.de/$\sim$xsc/).
Michael J. Mossinghoff, Polynomials with restricted coefficients and prescribed noncyclotomic factors, LMS J. Comput. Math. 6 (2003), 314–325. MR 2051588, DOI 10.1112/S1461157000000474
A. M. Odlyzko and B. Poonen, Zeros of polynomials with $0,1$ coefficients, Enseign. Math. (2) 39 (1993), no. 3-4, 317–348. MR 1252071
Andrzej Schinzel, On the reduced length of a polynomial with real coefficients, Funct. Approx. Comment. Math. 35 (2006), 271–306. MR 2271619, DOI 10.7169/facm/1229442629
GMP, The GNU Multiple Precision Arithmetic Library (available online at http://swox. com/gmp/).