Description of polygonal regions by polynomials of bounded degree

We show that every (possibly unbounded) convex polygon P in R-2 with in edges can be represented by inequalities p(1) > 0,...,pn >= 0, where the pi's are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m, k) <= n(m, k) <= (1 + epsilon(m))s(m, k) with s(m, k) := max{m/k, log(2) m} and epsilon(m) -> 0 as m -> infinity. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p(1) > 0,..., p(n) > 0 is also given. For k <= m / log(2) m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k.