Berline-Vergne赋值与广义置换面
Berline-Vergne valuation and generalized permutohedra
摘要：
推广De Loera等人的猜想，我们猜想整广义置换面都有正的Ehrhart系数。Berline和Vergne构造了一个赋值给多面体的面，它提供了一种将多面体的Ehrhart系数写为这些值的正和的方法。在已有结果的基础上，我们提出了一个更强的猜想：Berline-Vergne的赋值在置换面体上总是正的，这暗示了我们的第一个猜想。本文证明了我们关于Berline-Vergne赋值的强猜想对于维数不超过6是正确的，并且如果我们限制到余维数不超过3的面也是正确的。除了研究其正性结构外，我们还研究了Berline-Vergne赋值，并证明了它是在一定对称性约束下描述置换面体格点个数的McMullen公式的唯一构造.我们还给出了强猜想在混合赋值下的等价性说明。
英文摘要：
Generalizing a conjecture by De Loera et al., we conjecture that integral generalized permutohedra all have positive Ehrhart coefficients. Berline and Vergne construct a valuation that assigns values to faces of polytopes, which provides a way to write Ehrhart coefficients of a polytope as positive sums of these values. Based on available results, we pose a stronger conjecture: Berline-Vergne's valuation is always positive on permutohedra, which implies our first conjecture. This article proves that our strong conjecture on Berline-Vergne's valuation is true for dimension up to 6, and is true if we restrict to faces of codimension up to 3. In addition to investigating the positivity conjectures, we study the Berline-Vergne's valuation, and show that it is the unique construction for McMullen's formula used to describe number of lattice points in permutohedra under certain symmetry constraints. We also give an equivalent statement to the strong conjecture in terms of mixed valuations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
