Fuchsov's special point

In the theory of differential equations with complex time, the point $t=t_{0}\in \mathbb {C}$ ${\ displaystyle t = t_ {0} \ in \ mathbb {C}}$ ${\ displaystyle t = t_ {0} \ in \ mathbb {C}}$ called the Fuchsian singular point of the linear differential equation

{\dot {z}}=A(t)z,\quad z\in \mathbb {C} ^{n},\quad t\in \mathbb {C} ,

{\ displaystyle {\ dot {z}} = A (t) z, \ quad z \ in \ mathbb {C} ^ {n}, \ quad t \ in \ mathbb {C},}

{\ displaystyle {\ dot {z}} = A (t) z, \ quad z \ in \ mathbb {C} ^ {n}, \ quad t \ in \ mathbb {C},}

if the matrix of the system A (t) has a first-order pole in it. This is the simplest possible feature of a linear differential equation with complex time.

They also say that $t=\infty$ ${\ displaystyle t = \ infty}$ $t = \ infty$ is a Fuchsian singular point if the point $s=0$ ${\ displaystyle s = 0}$ $s = 0$ turns out to be fuchsian after replacement $t=1/s$ ${\ displaystyle t = 1 / s}$ ${\ displaystyle t = 1 / s}$ , in other words, if the matrix of the system $A(t)$ ${\ displaystyle A (t)}$ $A (t)$ tends to zero at infinity.

The simplest example

One-dimensional differential equation ${\dot {z}}={\frac {a}{t}}z$ ${\ displaystyle {\ dot {z}} = {\ frac {a} {t}} z}$ ${\dot {z}}={\frac {a}{t}}z$ has a Fuchsian singular point at zero, and its solutions are (generally speaking, multi-valued ) functions $z(t)=C\cdot t^{a}$ ${\ displaystyle z (t) = C \ cdot t ^ {a}}$ $z(t)=C\cdot t^{a}$ . When going around zero, the solution is multiplied by $\lambda =e^{2\pi ia}$ ${\ displaystyle \ lambda = e ^ {2 \ pi ia}}$ $\lambda =e^{2\pi ia}$ .

Growth of decisions and monodromy mapping

When approaching the Fuchsian singular point in any sector, the norm of the solution does not grow faster than polynomially:

\|z(t-t_{0})\|\leq C\|t-t_{0}\|^{-N}

{\ displaystyle \ | z (t-t_ {0}) \ | \ leq C \ | t-t_ {0} \ | ^ {- N}}

\|z(t-t_{0})\|\leq C\|t-t_{0}\|^{-N}

for some constants $C$ ${\ displaystyle C}$ $C$ and $N$ ${\ displaystyle N}$ $N$ . Thus, every Fuchsian singular point is regular .

Normal Poincare-Dulac-Levell Form

Hilbert's 21st Problem

Hilbert's twenty-first problem was that for given points on the Riemann sphere and a representation of the fundamental group of the complement to them, construct a system of differential equations with Fuchsian singularities at these points for which the monodromy turns out to be a given representation. For a long time it was believed that this problem was positively solved by Plemel (who published the solution in 1908 ), however, a mistake was discovered in his solution in the 1970s by Yu. S. Ilyashenko . In fact, the Plemelj construction made it possible to construct the required system with diagonalizability of at least one of the monodromy matrices. ^[one]

In 1989, A. A. Bolibrukh published ^[2] an example of a set of singular points and monodromy matrices, which cannot be realized by any Fuchsian system - thereby negatively solving the problem.

Literature

↑ Yu. S. Ilyashenko, “The nonlinear Riemann-Hilbert problem ”, Differential equations with real and complex time, Collected papers, Tr. MIAN, 213, Science, M., 1997, p. 10-34.
↑ A. A. Bolibruch, “The Riemann-Hilbert problem on the complex projective line” , Mat. Notes, 46: 3 (1989), 118-120

A. A. Bolibrukh, Inverse Monodromy Problems in the Analytical Theory of Differential Equations, M.: MCCNMO, 2009.
Yu. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, AMS, 2007.

[1] Yu. S. Ilyashenko, “The nonlinear Riemann-Hilbert problem ”, Differential equations with real and complex time, Collected papers, Tr. MIAN, 213, Science, M., 1997, p. 10-34.

[2] A. A. Bolibruch, “The Riemann-Hilbert problem on the complex projective line” , Mat. Notes, 46: 3 (1989), 118-120