On the absolute Mahler measure of polynomials having all zeros in a sector. (English) Zbl 0820.11064

Let \(P(z)\neq z\) be a monic polynomial with integer coefficients, irreducible over the rationals, of degree \(d\geq 1\), and having zeros \(\alpha_ 1, \dots, \alpha_ d\). The absolute Mahler measure of \(P\) is defined by \(\Omega (P)= \prod_{i=1}^ d \max(1, | \alpha_ i |^{1/d})\). A result of M. Langevin [C. R. Acad. Sci., Paris, Sér. I 303, 523–526 (1986; Zbl 0604.12001)] implies that, for \(0\leq \theta< \pi\), there is a constant \(c(\theta) >1\) such that, if all zeros of \(P\) lie in the sector \(\{z: | \arg z| \leq \theta\}\), and \(P\) is not cyclotomic, then \(\Omega (P)\geq c(\theta)\).
The aim of this paper is to find the best possible (largest) value of \(c(\theta)\) for as many values of \(\theta\) as possible. The authors give explicit upper and lower bounds for \(c(\theta)\). These bounds coincide for nine intervals \([\theta_ i, \theta'_ i]\) for which \(c(\theta)\) is therefore known exactly. The function \(c(\theta)\) is shown to be discontinuous at the left endpoints of these intervals. The angle \(\pi/2\) lies in one of these intervals giving the interesting result that if \(P\) is a monic irreducible polynomial with integer coefficients such that all its zeros have positive real part then either \(P(z)= z-1\) or \(P(z)= z^ 2- z+1\) or \(\Omega (P)\geq 1.12933793 \dots\), where the latter is \(\Omega (z^ 6- 2z^ 5+ 4z^ 4- 5z^ 3+ 4z^ 2- 2z+ 1)\), so the estimate is best possible. The corresponding \(\alpha\) satisfies \(\alpha+ 1/\alpha= \theta_ 0^ 2\), where \(\theta_ 0= 1.3247 \dots\) is the smallest Pisot number.


11R09 Polynomials (irreducibility, etc.)
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)


Zbl 0604.12001
Full Text: DOI


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