AI 中文总结
本文推广了永久式和混合判别式在双随机输入下的唯一极小化结果至近双随机边际，并将实稳定多项式的唯一极小化扩展至强对数凹多项式，同时讨论了相关开放问题。
AI 中文摘要
正矩阵的永久式和混合判别式是经典问题，我们对其精确计算不期望存在高效算法。因此，大量工作致力于理解如何对这些量进行界定和近似计算。该领域的一条研究路线始于第一作者的结果，其中通过稳定多项式的简单证明，证明了双随机输入下这两个问题的 van der Waerden 下界 $\ rac{n!}{n^n}$。除了下界本身，相同的技术还用于证明永久式和混合判别式在某个自然的对称输入处取得唯一最小值。本文从两个方面推广了这些结果。首先，我们将唯一极小化结果从双随机输入扩展到接近双随机的其他边际。这为混合判别式提供了双随机情形之外的首个此类唯一极小化结果。我们还讨论了为何不能期望类似结果对所有边际普遍成立。其次，我们将实稳定多项式的唯一极小化结果扩展到双随机情形下的强对数凹（即洛伦兹）多项式。这对应了先前关于混合体积唯一极小化的类似结果。最后，我们讨论了与这些结果相关的各种开放问题。
英文摘要
The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of $\frac{n!}{n^n}$ are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.
