Auxiliary polynomials for some problems regarding Mahler's measure

Dubickas, Artūras; Mossinghoff, Michael J.

doi:10.4064/aa119-1-5

Cited by 16 publications

(34 citation statements)

References 11 publications

(18 reference statements)

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“…In [5], Lehmer's conjecture was confirmed for polynomials with odd coefficients, so, in particular, for Littlewood polynomials. See also [10] for better numerical estimates.…”

Section: Artūras Dubickas and Jonas Jankauskasmentioning

confidence: 99%

On Newman polynomials which divide no Littlewood polynomial

Dubickas

Jankauskas

2009

Math. Comp.

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Abstract. Recall that a polynomial P (x) ∈ 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 α 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 ∈ {−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 N , Littlewood polynomials V L and {−1, 0, 1} polynomials V are distinct in the sense that between them there are only trivial relationsThe proofs of several main results (after some preparation) are computational.

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“…In [5], Lehmer's conjecture was confirmed for polynomials with odd coefficients, so, in particular, for Littlewood polynomials. See also [10] for better numerical estimates.…”

Section: Artūras Dubickas and Jonas Jankauskasmentioning

confidence: 99%

On Newman polynomials which divide no Littlewood polynomial

Dubickas

Jankauskas

2009

Math. Comp.

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“…if m ≥ 4 even (the even case having already been obtained and improved in [2]). We remark also that the same computation that yields the value of c 3 in Theorem 1.1 immediately produces the lower bound…”

Section: Introductionmentioning

confidence: 99%

“…where d is the degree of the noncyclotomic part of f (the type of bound obtained in Theorem 2.2 of [2]). We note the easily obtained (if asymptotically less precise) bound…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Heights of roots of polynomials with odd coefficients

Garza¹,

Ishak²,

Mossinghoff³

et al. 2010

J. Théor. Nombres Bordeaux

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“…Many particular cases of this conjecture have been solved, for instance if the minimal polynomial of x is non-reciprocal (see Smyth [35]) or has odd coefficients (see Borwein, Dobrowolski and Mossinghoff [5], or Dubickas and Mossinghoff [10] for further results).…”

Section: Introductionmentioning

confidence: 99%