Matrix coefficient realization theory of noncommutative rational functions.
Matrix coefficient realization theory of noncommutative rational functions.
复制标题
非交换有理函数的矩阵系数实现理论。
DOI：
10.1016/j.jalgebra.2017.12.009
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Jurij Volčič
中科院分区：
文献类型：
--
作者：
Jurij Volčič
非交换有理函数，即，自由代数的分数的泛斜域的元素，可以通过对矩阵元组的非交换有理表达式的求值来定义。这种解释扩展了他们传统上的重要作用，理论的分工环，并引起他们的应用在其他领域，从自由真实的代数几何系统和控制理论。如果一个非交换有理函数在原点是正则的，它可以用一个线性对象来描述，称为实化。本文给出了实现理论的一个推广，它适用于任意非交换有理函数，并且非常适合于研究矩阵求值，特别值得注意的是极小实现，它弥补了非交换有理函数缺乏标准形的缺陷。实现的非极小性是由与之相关的障碍模块来评估的，它们使我们能够设计出一种有效的方法来获得最小的实现。用它们来描述非交换有理函数的稳定扩张域，并定义一个数值不变量来度量其复杂性。利用这些结果，我们确定了具体的合理身份测试的大小界限，构造最小的对称实现，并证明了一个有效的局部-整体原则的线性依赖的非交换有理函数。
Noncommutative rational functions, i.e., elements of the universal skew field of fractions of a free algebra, can be defined through evaluations of noncommutative rational expressions on tuples of matrices. This interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas, from free real algebraic geometry to systems and control theory. If a noncommutative rational function is regular at the origin, it can be described by a linear object, called arealization. In this article we present an extension of the realization theory that is applicable toarbitrarynoncommutative rational functions and is well-adapted for studying matrix evaluations.Of special interest are the minimal realizations, which compensate the absence of a canonical form for noncommutative rational functions. The non-minimality of a realization is assessed by obstruction modules associated with it; they enable us to devise an efficient method for obtaining minimal realizations. With them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity. Using these results we determine concrete size bounds for rational identity testing, construct minimal symmetric realizations and prove an effective local–global principle for linear dependence of noncommutative rational functions.