# Piecemeal remarks on rational matrix functions of a complex variable

The global theory of rational linear systems on $\mathbb C P^1$ requires the study of (rational) gauge transformations which are holomorphic and holomorphically invertible except for a single point. If this point is at infinity, then the matrix of such transformation is necessarily polynomial with constant nonzero determinant. Such matrix functions are provisionally referred to as monopoles, $H(t)\in\textrm{GL}(n,\mathbb C[t]),\ \text{det}H=\text{const}\ne 0$.

Multiplication of a rational matrix function $H(t)$ from the left by a monopole matrix $\begin{pmatrix}1 & t\\ & 1\end{pmatrix}$ corresponds to adding the second row of $H$, multiplied by $t$, to the first row. Thus manipulations with rows of $H$, which aim at Gauss-type elimination of certain monomials from matrix elements, can be represented as gauge actions of the monopole group. The principal result that will be used throughout the next few lectures, is the following Bolibruch Permutation Lemma.

Lemma. Let $H(t)$ be the germ of a matrix function, holomorphic and invertible at $t=\infty$. Then for any ordered tuple of integer numbers $D=\{d_1,\dots,d_n\}$ the product $t^D\,H(t)$, $t^D=\text{diag}(t^{d_1},\dots,t^{d_n})$, is monopole equivalent to a product of the form $H'(t)\,t^{D'}$, where $H'(t)$ is also holomorphic and invertible at $t=\infty$, and $D'$ is a permutation of the tuple $D$.

The proof of this result is not difficult, yet is too technical to be delivered in the classroom.