A set $\mathcal{V}$ in the tridisk $\mathbb{D}^3$ has the polynomial extension property if for every polynomial $p$ there is a function $\phi$ on $\mathbb{D}^3$ so that $\| \phi \|_{\mathbb{D}^3} = \| p \|_{\mathcal{V}}$ and $\phi |_{\mathcal{V}} = p|_{\mathcal{V}}$. We study sets $\mathcal{V}$ that are relatively polynomially convex and have the polynomial extension property. If $\mathcal{V}$ is one-dimensional, and is either algebraic, or has polynomially convex projections, we show that it is a retract. If $\mathcal{V}$ is two-dimensional, we show that either it is a retract, or, for any choice of the coordinate functions, it is the graph of a function of two variables.

中文翻译：

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Tridisk上有界全纯函数的扩展

如果每个多项式$ p $在$ \ mathbb {D} ^ 3上有函数$ \ phi $，则三元盘$ \ mathbb {D} ^ 3 $中的集合$ \ mathcal {V} $具有多项式扩展属性。 $使$ \ | \ phi \ | _ {\ mathbb {D} ^ 3} = \ | p \ | _ {\ mathcal {V}} $和$ \ phi | _ {\ mathcal {V}} = p | _ {\ mathcal {V}} $。我们研究集合\\ mathcal {V} $，它们是相对多项式凸的，并且具有多项式扩展性质。如果$ \ mathcal {V} $是一维的，或者是代数的，或者具有多项式凸投影，则表明它是一个缩进。如果$ \ mathcal {V} $是二维的，则表明它是缩进的，或者对于任何坐标函数选择，它都是两个变量的函数的图。