The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

中文翻译：

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通过容量保持算子计数匹配

多项式容量的概念是由 Gurvits 于 2005 年左右引入的，最初是为了对双随机矩阵的永久项的 van der Waerden 下界和规则二部图的完美匹配的 Schrijver 不等式给出极大简化的证明。自从这项开创性的工作以来，容量的概念已被用于限制各种组合量并给出多项式时间算法来近似这些量（例如拟阵的基数）。这些类型的结果通常通过给出特定微分算子可以改变给定多项式容量的范围来证明。在本文中，我们通过为所有非退化真实稳定性保持器提供严格的容量保持范围来统一围绕这种容量保持算子的理论。