  We present a generic approach that allows us to develop a fully
polynomial-time approximation scheme (FTPAS) for minimizing nonlinear functions
over the integer points in a rational polyhedron in fixed dimension. The
approach combines the subdivision strategy of Papadimitriou and Yannakakis
(2000) with ideas similar to those commonly used to derive real algebraic
certificates of positivity for polynomials. Our general approach is widely
applicable. We apply it, for instance, to the Motzkin polynomial and to
indefinite quadratic forms $x^T Q x$ in a fixed number of variables, where $Q$
has at most one positive, or at most one negative eigenvalue. In dimension
three, this leads to an FPTAS for general $Q$.
