Littlewood Pisot numbers. (English) Zbl 1095.11049
The author studies Littlewood Pisot numbers which are defined as Pisot numbers whose minimal polynomials are Littlewood polynomials, i.e., those with \(\pm 1\) coefficients. He also defines Littlewood Salem numbers as Salem numbers which are roots of (not necessarily irreducible) Littlewood polynomials.
The main theorem of this paper asserts that every Littlewood Pisot number has the minimal polynomial \(x^n-x^{n-1}-x^{n-2}-\dots-x-1=0,\) where \(n \geq 2\). The proof involves Boyd’s algorithm for determining all Pisot numbers in a given interval.
The author also proves that every Littlewood Pisot number is a limit point of Littlewood Salem numbers from below. The proof uses some arguments of Borwein, Dobrowolski and Mossinghoff. They earlier proved this for the golden mean \((1+\sqrt{5})/2\) which is the smallest Littlewood Pisot number. Finally, he proves a result which implies that every reciprocal Littlewood polynomial of odd degree has at least three unimodular roots.
The main theorem of this paper asserts that every Littlewood Pisot number has the minimal polynomial \(x^n-x^{n-1}-x^{n-2}-\dots-x-1=0,\) where \(n \geq 2\). The proof involves Boyd’s algorithm for determining all Pisot numbers in a given interval.
The author also proves that every Littlewood Pisot number is a limit point of Littlewood Salem numbers from below. The proof uses some arguments of Borwein, Dobrowolski and Mossinghoff. They earlier proved this for the golden mean \((1+\sqrt{5})/2\) which is the smallest Littlewood Pisot number. Finally, he proves a result which implies that every reciprocal Littlewood polynomial of odd degree has at least three unimodular roots.
Reviewer: Artūras Dubickas (Vilnius)
MSC:
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11R09 | Polynomials (irreducibility, etc.) |
| 11C08 | Polynomials in number theory |
References:
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| [2] | Borwein, P., Computational Excursions in Analysis and Number Theory, (CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (2002), Springer: Springer Berlin), 15-16 · Zbl 1020.12001 |
| [4] | Borwein, P.; Hare, K. G.; Mossinghoff, M. J., The Mahler measure of polynomials with odd coefficients, Bull. London Math. Soc., 36, 332-338 (2004) · Zbl 1143.11349 |
| [5] | Boyd, D. W., Pisot and Salem numbers in intervals of the real line, Math. Comp., 32, 1244-1260 (1978) · Zbl 0395.12004 |
| [6] | Dufresnoy, J.; Pisot, Ch., Étude de certaines fonctions méromorphes bornées sur le cercle unité, application à un ensemble fermé d’entiers algébriques, Ann. Sci. École. Norm. Sup., 72, 3, 69-92 (1955) · Zbl 0064.03703 |
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| [11] | Salem, R., Algebraic Numbers and Fourier Analysis (1963), Heath Mathematical Monographs · Zbl 0126.07802 |
| [12] | Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic number, Bull. London Math. Soc., 3, 169-175 (1971) · Zbl 0235.12003 |
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