For each m in N and each sufficiently large d in 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-ε, this estimate is better than the estimate m^d(1+ε) 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.

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