Speaker: Christophe Doche
Title: Mahler measure and Zhang-Zagier height
Carslaw 273, Friday 24 September
4:05-4:55PM

Given a monic polynomial P with coefficients in Z, the mahler measure 
of P is equal to the product of the absolute value of the the roots 
of P which are outside the unit disk. Lehmer's problem asks if 1 is a 
limit point of the set containing all the mahler measures of all the 
polynomials with integer coefficients.
Even if the problem is still open, it is possible to conclude for some
families of polynomials, or equivalenlty for some families of algebraic 
numbers.  The object of this talk is to strengten previous work of 
Zhang, Beukers and Zagier which gives an answer to Lehmer's problem 
for such a special family.  Indeed, let h(alpha) be the height of 
alpha, that is the normazized Mahler measure of its minimal polynomial 
and ZZ(alpha) be h(alpha)h(1-alpha).  In this talk we improve the lower 
bound found by Zagier, namely ZZ(alpha) >= sqrt((1+sqrt(5))/2) except 
for finitely many exceptions.  We also give an algorithm to find 
algebraic numbers of small Zhang-Zagier height and we give limit 
points of the corresponding spectrum.