Optimization of a convex program with a polynomial perturbation

We consider the problem of minimizing a convex function plus a polynomial p over a convex body K. We give an algorithm that outputs a solution x whose value is within rangeK(p) of the optimum value, where rangeK(p) = supx∈K p(x)− infx∈K p(x). When p depends only on a constant number of variables, the algorithm runs in time polynomial in 1/ , the degree of p, the time to round K and the time to solve the convex program that results by setting p = 0.