论文标题
仿射压力：Unity和Kalai的重建猜想的分区
Affine stresses: the partition of unity and Kalai's reconstruction conjectures
论文作者
论文摘要
卡莱（Kalai）猜想，如果$ p $是一个简单的$ d $ - polytope，没有尺寸$ d-1 $的面孔，则$ p $的图和affine $ 2 $ 2 $ p $ $ p $ p $ p $ p $ p $ p $的空间。 We propose a higher-dimensional generalization of this conjecture: if $2\leq i\leq d/2$ and $P$ is a simplicial $d$-polytope that has no missing faces of dimension $\geq d-i+1$, then the space of affine $i$-stresses of $P$ determines the space of affine $1$-stresses of $P$.我们证明了（1）$ k $的$ d $ -d $ - $ 2 \ leq i \ leq i \ leq k \ leq k \ leq d/2-1 $，（2）$ d $ - 没有尺寸的面孔$ \ geq d-2 $ 2 $ 2 $（3）$（3）$（3）$ - 2 $ - d-d-d-d-emp emq（d-d-d-d-emp eermq）我\ leq d/2 $）。我们还讨论了几个相关的结果和猜想。例如，我们表明，如果$ p $是一个简单的$ d $ - 多型物种，没有尺寸$ \ geq d-2i+2 $的面孔，则是$ p $的$（i-1）$ - 骨架，以及一组affine $ i $ i $ p $ $ p $的符号向量，确定$ p $ $ p $ $ p $。 Along the way, we establish the partition of unity of affine stresses: for any $1\leq i\leq (d-1)/2$, the space of affine $i$-stresses of a simplicial $d$-polytope as well as the space of affine $i$-stresses of a simplicial $(d-1)$-sphere (with a generic embedding) can be expressed as the sum of affine $i$-stress顶点星的空间。这类似于Adiprasito的Cohen--Macaulay复合物的线性应力统一分区。
Kalai conjectured that if $P$ is a simplicial $d$-polytope that has no missing faces of dimension $d-1$, then the graph of $P$ and the space of affine $2$-stresses of $P$ determine $P$ up to affine equivalence. We propose a higher-dimensional generalization of this conjecture: if $2\leq i\leq d/2$ and $P$ is a simplicial $d$-polytope that has no missing faces of dimension $\geq d-i+1$, then the space of affine $i$-stresses of $P$ determines the space of affine $1$-stresses of $P$. We prove this conjecture for (1) $k$-stacked $d$-polytopes with $2\leq i\leq k\leq d/2-1$, (2) $d$-polytopes that have no missing faces of dimension $\geq d-2i+2$, and (3) flag PL $(d-1)$-spheres with generic embeddings (for all $2\leq i\leq d/2$). We also discuss several related results and conjectures. For instance, we show that if $P$ is a simplicial $d$-polytope that has no missing faces of dimension $\geq d-2i+2$, then the $(i-1)$-skeleton of $P$ and the set of sign vectors of affine $i$-stresses of $P$ determine the combinatorial type of $P$. Along the way, we establish the partition of unity of affine stresses: for any $1\leq i\leq (d-1)/2$, the space of affine $i$-stresses of a simplicial $d$-polytope as well as the space of affine $i$-stresses of a simplicial $(d-1)$-sphere (with a generic embedding) can be expressed as the sum of affine $i$-stress spaces of vertex stars. This is analogous to Adiprasito's partition of unity of linear stresses for Cohen--Macaulay complexes.
