The following case of integer polynomial optimization is considered: Minimize a polynomial F̂ on the set of integer points described by an inequality system F1 ≤ 0, . . . , Fs ≤ 0, where F̂ , F1, . . . , Fs are quasiconvex polynomials in n variables with integer coefficients. An algorithm solving this problem is designed that belongs to the time-complexity class O(s) · lO(1) ·dO(n) ·2O(n3), where d ≥ 2 is an upper bound for the total degree of the polynomials involved and l denotes the maximum binary length of all coefficients. The algorithm is polynomial for a fixed number n of variables and represents a direct generalization of H.W.Lenstra’s algorithm  in integer linear optimization. Our complexity–result improves as well the one due to Bank et al.  as the one due to Khachiyan and Porkolab  for integer polynomial optimization in the considered case.