  We consider the problem of computing the partition function $\sum_x
e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ is a quadratic or
cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic
polynomial $f$, we show that the partition function can be approximated within
relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln
\epsilon)}$ time if the Lipschitz constant of the non-linear part of $f$ with
respect to the $\ell^1$ metric on the Boolean cube does not exceed $1-\delta$,
for any $\delta >0$, fixed in advance. For a cubic polynomial $f$, we get the
same result under a somewhat stronger condition.
  We apply the method of polynomial interpolation, for which we prove that
$\sum_x e^{\tilde{f}(x)} \ne 0$ for complex-valued polynomials $\tilde{f}$ in a
neighborhood of a real-valued $f$ satisfying the above mentioned conditions.
The bounds are asymptotically optimal. Results on the zero-free region are
interpreted as the absence of a phase transition in the Lee - Yang sense in the
corresponding Ising model. The novel feature of the bounds is that they control
the total interaction of each vertex but not every single interaction of sets
of vertices.
