Wednesday, December 3, 2008

IH16 and friends: the final dash

Filed under: links, research announcement — Sergei Yakovenko @ 11:03
Tags: Abelian integrals, global theory of Fuchsian systems, Hilbert 16th problem, isomonodromic deformations

Finally the two texts concerned with solution of the Infinitesimal Hilbert problem, are put into the polished form (including the publisher’s LaTeX style files). The new revisions, already uploaded to ArXiv, differ from the initial submissions only by corrected typos, a few rearrangements aimed at improving the readability of the texts, and a couple of more references added. There is absolutely no need to read the new revision if you already have read the first one.

Mostly for the reasons of “internal convenience” the complete references are repoduced here:

G. Binyamini and S. Yakovenko, Polynomial Bounds for Oscillation of Solutions of Fuchsian Systems, posted as arXiv:0808.2950v2 [math.DS], 36 p.p., submitted to Ann. Inst. Fourier (Dec. 2008), accepted (February, 2009)
G. Binyamini, D. Novikov and S. Yakovenko, On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem, posted as arXiv:0808.2952v2 [math.DS], 57 p.p., submitted to Inventiones Mathematicae (Nov. 2008), accepted (Oct. 2009).

Wednesday, March 19, 2008

Lecture 4 (Thu, Mar 13, 2008)

Filed under: Analytic ODE course, lecture — Sergei Yakovenko @ 9:04
Tags: gauge equivalence, global theory of Fuchsian systems, matrix functions, monopole

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 $D$ .