A PTAS for the minimization of polynomials of fixed degree over the simplex

We consider the problem of computing the minimum value pmin taken by a polynomial p(x) of degree d over the standard simplex Δ. This is an NP-hard problem already for degree d=2. For any integer k⩾1, by minimizing p(x) over the set of rational points in Δ with denominator k, one obtains a hierarchy of upper bounds pΔ(k) converging to pmin as k⟶∞. These upper approximations are intimately linked to a hierarchy of lower bounds for pmin constructed via Pólya's theorem about representations of positive forms on the simplex. Revisiting the proof of Pólya's theorem allows us to give estimates on the quality of these upper and lower approximations for pmin. Moreover, we show that the bounds pΔ(k) yield a polynomial time approximation scheme for the minimization of polynomials of fixed degree d on the simplex, extending an earlier result of Bomze and De Klerk for degree d=2.