On sum of squares certificates of non-negativity on a strip

In [6], Murray Marshall proved that every f∈R[X,Y] non-negative on the strip [0,1]×R can be written as f=σ0+σ1X(1−X) with σ0,σ1 sums of squares in R[X,Y]. this work, we present a few results concerning representation particular cases. First, under assumption degY⁡f≤2, by characterizing extreme rays suitable cone, obtain degree bound for each term. Then, consider case f positive and non-vanishing at infinity, show again term, coming from constructive method to sum representation. Finally, also works having only finite number zeros, all them lying boundary strip, such ∂f∂X does not vanish any them.