Description of polygonal regions by polynomials of bounded degree

We show that every (possibly unbounded) convex polygon P in ({mathbb{R}^2}) with m edges can be represented by inequalities p ₁ ≥ 0, . . ., p _n ≥ 0, where the p _i ’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) leq n(m, k) leq (1+varepsilon_m) s(m, k)}) with s(m,k) ? max {m/k, log₂ m} and ({varepsilon_m rightarrow 0}) as ({m rightarrow infty}). 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/log₂ m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k.