Polynomials non-negative on strips and half-strips

In 2008, Marshall (2010) [4] settled a long-standing open problem by showing that if f(x,y)∈R[x,y] is a polynomial that is non-negative on the strip [0,1]×R, then there exist sums of squares σ(x,y),τ(x,y)∈∑R[x,y]2 such that f(x,y)=σ(x,y)+τ(x,y)(x−x2). In this paper, we generalize Marshall’s result to various strips and half-strips in the plane. Our results give many new examples of non-compact semialgebraic sets in R2 for which one can characterize all polynomials which are non-negative on the set. For example, we show that if U is a compact subset of the real line and {g1,…,gk} a specific set of generators for U as a semialgebraic set, then whenever f(x,y) is non-negative on U×R, there are sums of squares s0,…,sk such that f=s0+s1g1+⋯+skgk.