Van der Waerden/Schrijver-Valiant类猜想与稳定（亦称双曲）齐次多项式：一个定理
Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials : One Theorem for all
摘要：
令$p$是$n$变量$p(z_1,...,z_n) = p(Z)$、$Z \in C^{n}$中的次数$n$的齐次多项式。如果$p(z_1,...,z_n) \neq 0$提供了实部$Re(z_i) > 0, 1 \leq i \leq n$，我们称这样的多项式为$p$ { \bf H-稳定}。这个来自{PDMATHAMT控制理论}的概念与{PDMATHAMT PDE}理论中密集使用的{PDMATHAMT双曲性}概念密切相关。本文的主要定理是：如果$p(x_1,...,x_n)$是一个具有非负系数的齐次{ \bf H-Stable} $n$多项式; $deg_{p}(i)$是变量$x_i$的最大程度，那么下面的不等式成立$C_i = \min(deg_{p}(i),i)$这个不等式是一个巨大的（和统一）推广的范德瓦尔登猜想的永久性的双重随机矩阵以及Schrijver-关于$k$ -正则二部图中完美匹配数的Valiant猜想。这两个著名的结果对应于{ \bf H-稳定}多项式是产品的线性形式。我们的证明是相对简单和“非计算性的”;它只使用了复数和AM/GM不等式的非常基本的性质。
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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