论文标题
在埃尔米特的某些代数特性上
On Some Algebraic Properties of Hermite--Padé Polynomials
论文作者
论文摘要
令$ [f_0，\ dots，f_m] $为$ 1/z $，$ f_j（\ infty）\ neq0 $的非负功率的系列元组。据认为元组处于“一般位置”。我们将I型和II型Hermite-Padé多项式构造给给定的元组，分别为$ \ leq {n} $和$ \ leq {Mn} $以及相应的$（m+1）$ - 具有以下属性的多index。令$ m_1（z）$和$ m_2（z）$为两个$（m+1）\ times（m+1）$多项式矩阵，$ m_1（z），m_2（z）\ in \ in \ operatoratorname {gl}（gl}（m+1，m+1，\ mathb c [z]）$，由type i ii和type ii ii herm-pade产生。然后，我们有$ m_1（z）m_2（z）\ equiv i_ {m+1} $，其中$ i_ {m+1} $是身份$（m+1）\ times（m+1）$ - 矩阵。 结果是由Hermite-Padé多项式在研究微分方程系统的单一特性研究中的一些新颖应用的动机。
Let $[f_0,\dots,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty)\neq0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Padé polynomials to the given tuple of degrees $\leq{n}$ and $\leq{mn}$ respectively and the corresponding $(m+1)$-multi-indexes with the following property. Let $M_1(z)$ and $M_2(z)$ be two $(m+1)\times(m+1)$ polynomial matrices, $M_1(z),M_2(z)\in\operatorname{GL}(m+1,\mathbb C[z])$, generated by type I and type II Hermite--Padé polynomials respectively. Then we have $M_1(z)M_2(z)\equiv I_{m+1}$, where $I_{m+1}$ is the identity $(m+1)\times(m+1)$-matrix. The result is motivated by some novel applications of Hermite--Padé polynomials to the investigation of monodromy properties of Fuchsian systems of differential equations.
