On the number of polynomials of small house

Suppose that ɛ is a positive number, and letd be a sufficiently large positive integer. We consider a set of monic polynomials of degreed with integer coefficients and roots lying in the disk of radiusd ^ɛ/d. We prove that the number of such polynomials is less than exp(d ^2/3+ɛ). Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 214–219, April–June, 1999.