On an application of a method of Fekete-Szegö to Lehmer’s problem. (Méthode de Fekete-Szegö et problème de Lehmer.) (French) Zbl 0585.12013

Langevin, Michel

C. R. Acad. Sci., Paris, Sér. I 301, 463-466 (1985).

For a polynomial \(P\) with leading coefficient \(a_ 0\) and roots \(x_ 1,\ldots,x_ d\), define \[ M(P)=| a_ 0| \prod \max (| x_ i|,1), \] the Mahler measure of \(P\). Using the Fekete-Szegö theory of transfinite diameter, the author shows that if \(V\) is any open set containing a point of the unit circle there is a constant \(C(V)>1\) so that if \(P\) is any irreducible noncyclotomic polynomial with integer coefficients, of degree \(d\), none of whose roots lie in \(V\), then \(M(P)\geq C(V)^ d\).
It would be interesting to give a good estimate for \(C(V)\) when \(V\) is a small disk centered on the unit circle.

Reviewer: David W. Boyd (Vancouver)

MSC:

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

Keywords:

Lehmer’s problem; Mahler measure; transfinite diameter