Polynomials with the half-plane property and matroid theory
Polynomials with the half-plane property and matroid theory
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DOI：
10.1016/j.aim.2007.05.011
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发表时间：
2007-12-01
影响因子：
1.7
通讯作者：
Braenden, Petter
中科院分区：
文献类型：
--
作者：
Braenden, Petter
如果存在 C 的一个开半平面 H 子集，且其边界包含原点，且只要所有变量都在 H 中，则称多项式 f 具有半平面性质。本文回答了有关具有半平面性质的多元多项式与拟阵理论的几个开放性问题。(1) 我们证明了具有半平面性质的多元多项式的支持是一个跳跃系统。这回答了 Choe、Oxley、Sokal 和 Wagner 提出的一个开放性问题，并概括了他们最近的结果，声称只要多项式也是齐次的，情况也是如此。(2) 我们证明多元多元仿射多项式 f 是 R[z(1),..., z(n)] 的元素，具有半平面性质（相对于上半平面）当且仅当偏导数 f/偏导数 (zi)(x) 中心点偏导数 f/偏导数(zj)(x)-偏导数(2)f/偏导数(zi)偏导数(zj)(x)中心点 f(x)>= 0 for all x is an element of R-n and 1
A polynomial f is said to have the half-plane property if there is an open half-plane H subset of C, whose boundary contains the origin, such that f is non-zero whenever all the variables are in H. This paper answers several open questions relating multivariate polynomials with the half-plane property to matroid theory.(1) We prove that the support of a multivariate polynomial with the half-plane property is a jump system. This answers an open question posed by Choe, Oxley, Sokal and Wagner and generalizes their recent result claiming that the same is true whenever the polynomial is also homogeneous.(2) We prove that a multivariate multi-affine polynomial f is an element of R[z(1),..., z(n)] has the half-plane property (with respect to the upper half-plane) if and only ifpartial derivative f/partial derivative(zi)(x)center dot partial derivative f/partial derivative(zj)(x)-partial derivative(2)f/partial derivative(zi)partial derivative(zj)(x)center dot f(x)>= 0for all x is an element of R-n and 1