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摘要翻译：
设$p$是$n$变量中的$n$次齐次多项式，$p(z_1,...,z_n)=p(Z)$,$Z\在C^{n}$中。如果$P(z_1,...,z_n)\neq0$提供了实部$re(z_i)>0,1\leqi\leqn$,我们称这样的多项式为$p${\bf H-稳定}。这个来自{it控制论}的概念与{it PDE}理论中密集使用的{it双曲性}的概念密切相关。本文的主要定理表明：如果$P(x_1,...,x_n)$是非负系数的齐次{bf H-稳定}多项式；$deg_{p}(i)$是变量$x_i$的最大度，$C_i=\min(deg_{p}(i)，i)$和$$Cap(p)=\inf_{x_i>0,1\leq i\leq n}\frac{p(x_1,...,x_n)}{x_1...x_n}$$则下面的不等式为$$\frac{\partial^n}{\partial x_1...\partial x_n}p(0,...,0)\geq Cap(p)\prod_{2\leq i\leq n}(\frac{c_i-1}{C_i}）这个不等式是关于双随机矩阵永久变量的Van der Waerden猜想和关于$k$-正则二部图中完美匹配个数的Schrijver-Valiant猜想的一个巨大的（和统一的）推广。这两个著名的结果对应于线性形式乘积的{bf H-稳定}多项式。我们的证明是相对简单的，“非计算性的”；它只使用复数的基本性质和AM/GM不等式。
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英文标题：
《Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka
Hyperbolic) Homogeneous Polynomials : One Theorem for all》
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作者：
Leonid Gurvits
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Combinatorics 组合学
分类描述：Discrete mathematics, graph theory, enumeration, combinatorial optimization, Ramsey theory, combinatorial game theory
离散数学，图论，计数，组合优化，拉姆齐理论，组合对策论
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z)$, $Z \in C^{n}$. We call such a polynomial $p$ {\bf H-Stable} if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This notion from {\it Control Theory} is closely related to the notion of {\it Hyperbolicity} used intensively in the {\it PDE} theory. The main theorem in this paper states that if $p(x_1,...,x_n)$ is a homogeneous {\bf H-Stable} polynomial of degree $n$ with nonnegative coefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i = \min(deg_{p}(i),i)$ and $$ Cap(p) = \inf_{x_i > 0, 1 \leq i \leq n} \frac{p(x_1,...,x_n)}{x_1 ... x_n} $$ then the following inequality holds $$ \frac{\partial^n}{\partial x_1... \partial x_n} p(0,...,0) \geq Cap(p) \prod_{2 \leq i \leq n} (\frac{C_i -1}{C_i})^{C_{i}-1}. $$ This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs. These two famous results correspond to the {\bf H-Stable} polynomials which are products of linear forms. Our proof is relatively simple and ``noncomputational''; it uses just very basic properties of complex numbers and the AM/GM inequality.
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PDF链接：
https://arxiv.org/pdf/0711.3496
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