Polynomials with integral Mahler measures

For each m is an element of N and each sufficiently large d is an element of N , we give an upper bound for the number of integer polynomials of degree d and Mahler's measure m . We show that there are at most exp (11( md ) (2 / 3) ( log( md )) (4/3) ) of such polynomials. For 'small' m , i.e. m < d( 1/2- is an element of) , this estimate is better than the estimate m (d (1+ is an element of)) that comes from a corresponding upper bound on the number of integer polynomials of degree d and Mahler's measure at most m . By the results of Zaitseva and Protasov, our estimate has applications in the theory of self-affine 2 -attractors. We also show that for each integer m >= 3 there is a constant c = c ( m ) > 0 such that the number of monic integer irreducible expanding polynomials of sufficiently degree d and constant coefficient m (and hence with Mahler's measure equal to m ) is at least cd (m - 1) .