A well-known result of Murray Marshall states that for every $f \in \mathbb{R} [X,Y]$ non-negative on the strip $[0,1] \times \mathbb{R}$ can be written as $f= \sigma_0 + \sigma_1 X(1-X)$ with $\sigma_0, \sigma_1$ sums of squares in $\mathbb{R} [X,Y]$. In this work, we present a few results concerning this representation in particular cases. First, under the assumption ${\rm deg}_Y f \leq 2$, by characterizing the extreme rays of a suitable cone, we obtain a degree bound for each term. Then, we consider the case of $f$ positive on $[0,1] \times \mathbb{R}$ and non-vanishing at infinity, and we show again a degree bound for each term, coming from a constructive method to obtain the sum of squares representation. Finally, we show that this constructive method also works in the case of $f$ having only a finite number of zeros, all of them lying on the boundary of the strip, and such that $\frac{\partial f}{\partial X}$ does not vanish at any of them.

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关于条带上非负性证明的平方和

Murray Marshall 的一个众所周知的结果指出，对于带 $[0,1] \times \mathbb{R}$ 上的每一个 $f \in \mathbb{R} [X,Y]$ 可以写成作为 $f= \sigma_0 + \sigma_1 X(1-X)$ 与 $\sigma_0, \sigma_1$ $\mathbb{R} [X,Y]$ 中的平方和。在这项工作中，我们在特定情况下介绍了有关这种表示的一些结果。首先，在 ${\rm deg}_Y f \leq 2$ 的假设下，通过表征合适锥体的极端光线，我们获得了每一项的度数界限。然后，我们考虑 $[0,1] \times \mathbb{R}$ 上的 $f$ 为正且在无穷远处不消失的情况，我们再次展示了每个项的度数界限，来自于构造方法获得平方和表示。最后，我们证明了这种构造方法也适用于 $f$ 只有有限数量的零的情况，