AI 中文总结
该研究针对双圆盘或双半平面上的二元Schur与Herglotz矩阵值有理函数，建立有限维实现理论，刻画对称实现并得到相关行列式表示，拓展了无限维实现的已有成果。
AI 中文摘要
我们针对在双圆盘或双半平面上具有压缩性或非负实部的二元有理函数，提出了有限维实现理论。研究表明，实现公式仅取决于对应的基础域，而四类函数之间的区分完全由施加在实现矩阵上的显式矩阵不等式来体现。这些结果为有理Schur--Agler函数和Herglotz--Agler函数提供了有限维实现，拓展了此前的无限维相关成果。我们进一步通过实现数据的埃尔米特酉对称性来刻画对称实现，从而在对称双半平面上得到实现定理。最后，我们获得了对称稳定多项式的行列式表示，进而得到对称双半平面上稳定多项式的行列式表示；对于实域上的有理函数，相应表示可使用实元素的矩阵。
英文摘要
We present a finite-dimensional realization theory for bivariate rational functions that are contractive or have nonnegative real part on the bidisc or on the bihalfplane. We show that the realization formula depends only on the underlying domain, while the distinction between the four resulting function classes is captured entirely by explicit matrix inequalities imposed on the realization matrices. These results provide finite-dimensional realizations for rational Schur--Agler and Herglotz--Agler functions, extending the previous infinite-dimensional results. We further characterize symmetric realizations by means of a Hermitian unitary symmetry of the realization data, yielding realization theorems on the symmetrized bihalfplane. Finally, we obtain determinantal representations for symmetric stable polynomials and, consequently, for stable polynomials on the symmetrized bihalfplane. For rational functions over the real field the respective representations can use matrices with real entries.
Comments19 pages + appendix
