Before revision:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq }  1-\delta for all i.
After revision:  We prove that for any \delta >0, fixed in advance, the value of \sum_x e^{f(x) can be approximated within relative error 0 < \epsilon < 1 is quasi-polynomial n^{O(\ln n - \ln \epsilon)} time , as long as \sum_j |a_{ij| \leq } 1-\delta , for any \delta >0, fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition.