A real stable extension of the Vamos matroid polynomial
A real stable extension of the Vamos matroid polynomial
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Vamos 拟阵多项式的实稳定推广
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Y. Youm
中科院分区：
文献类型：
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作者：
Sam Burton;C. Vinzant;Y. Youm
2004年，Choe，Oxley，Sokal和Wagner建立了Matroid和Multiaffine真实稳定多项式之间的紧密联系。最近，布兰登（Branden）使用了这一理论和来自Vamos Matroid的多项式来反驳广义的LAX猜想。在这里，我们提出了VAMOS Matroid的10元素扩展，并证明其基础生成多项式是真正的稳定（即Matroid具有半平面属性）。我们通过大量的平方计算和Wagner和Wei给出的实际稳定性标准来做到这一点。像Vamos Matroid一样，该基曲线在任何场上都无法表示，并且其基础生成多项式的功能不能写为具有阳性半菲尼特遗传形式的线性矩阵的决定因素。
In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Recently, Branden used this theory and a polynomial coming from the Vamos matroid to disprove the generalized Lax conjecture. Here we present a 10-element extension of the Vamos matroid and prove that its basis generating polynomial is real stable (i.e. that the matroid has the half-plane property). We do this via large sums of squares computations and a criterion for real stability given by Wagner and Wei. Like the Vamos matroid, this matroid is not representable over any field and no power of its basis generating polynomial can be written as the determinant of a linear matrix with positive semidefinite Hermitian forms.