Robertson in 1988 suggested a model for the realization space of a convex d‐dimensional polytope and an approach via the implicit function theorem, which—in the case of a full rank Jacobian—proves that the realization space is a manifold of dimension $\mathrm{NG}(P):=d(f_0+f_{d-1})-f_{0,d-1}$. This is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3), and Robertson claimed this to be true for all polytopes, Mnëv's (1986/1988) universality theorem implies that it is not true in general. Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper, we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes, the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count NG(P) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For an explicit example with property (1), we analyze the classical 24‐cell: We show that the realization space has at least dimension $\mathrm{NG}(C_4^{(24)})=48$, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.

中文翻译：

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多边形实现空间的维数

罗伯逊（Robertson）于1988年提出了凸d维多边形的实现空间模型和隐式函数定理的一种方法，对于全秩雅可比矩阵，它证明了实现空间是维的流形。$\mathrm{NG}（P）：=d（F_0+F_{d-\mathrm{1个}}）-F_{0，d-\mathrm{1个}}$。这是对变量数量减去在实现空间定义中使用的二次方程式数量给定的维数的自然猜测。尽管这确实适用于许多自然类别的多面体（包括简单和简单的多面体，以及所有维数最多为3的多面体），罗伯逊声称这对所有多面体都是正确的，但Mnëv（1986/1988）的普遍性定理意味着：总的来说，这是不正确的。实际上，（1）居中的实现空间通常不是平滑嵌入的流形，并且（2）它不具有尺寸NG（P） 大体。在本文中，我们为分析实现空间开发了雅可比准则。从这些中我们可以轻松地得出，对于各种大型自然类别的多面体，实现空间的确是流形的，其尺寸由NG（P）给出。但是，我们还确定了最小的多面体，其中维数为NG（P），因此罗伯逊的主张失败了，其中包括三棱锥上的双锥体。对于具有属性（1）的显式示例，我们分析经典的24单元格：我们证明实现空间至少具有维$\mathrm{NG}（C_4^{（24）}）=48$，并且在某些点上它是该尺寸的流形，但并不是到处都作为流形平滑地嵌入其中。