Determinantal representations of hyperbolic plane curves: An elementary approach
Determinantal representations of hyperbolic plane curves: An elementary approach
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双曲平面曲线的行列式表示：一种基本方法
DOI：
10.1016/j.jsc.2013.05.004
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
C. Vinzant
中科院分区：
文献类型：
--
作者：
D. Plaumann;C. Vinzant
在2007年，Helton和Vinnikov证明了每一个双曲平面曲线都有一个确定的真实的对称行列式表示。通过允许埃尔米特矩阵，我们能够给出一个新的证明，只依赖于基本的平面曲线的相交理论。我们表明，矩阵的线性形式是确定的当且仅当它的共同极大的未成年人交织其行列式和扩展的经典建设的行列式表示的狄克逊从1902年。像Helton-Vinnikov定理一样，这意味着平面上的每个双曲区域都由线性矩阵不等式定义。
In 2007, Helton and Vinnikov proved that every hyperbolic plane curve has a definite real symmetric determinantal representation. By allowing for Hermitian matrices instead, we are able to give a new proof that relies only on the basic intersection theory of plane curves. We show that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extend a classical construction of determinantal representations of Dixon from 1902. Like the Helton–Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality.