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**Gauss-Manin systems of polynomials of two variables can be made Fuchsian.**
*(English)*
Zbl 0979.34066

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1-10, 1999. Sofia: Coral Press Scientific Publishing. 105-126 (2000).

Consider a system of linear equations \(dx/dt= A(t)x\) (\(A\) is a matrix meromorphic in the projective line), which is regular-singular at each pole \(a_j\) of \(A\) and at infinity. Such a system is called Fuchsian if \(A\) is of the form \(\sum A_j/(t- a_j)\) with constant \(A_j\). Here, the following Riemann-Hilbert-type theorem is proved modulo a conjecture due to A. A. Bolibrukh: If at every \(a_j\) local monodromy in its Jordan form has only Jordan blocks of size \(\leq 2\), then the monodromy group is realizable by a Fuchsian system for any prescribed set of poles.

The theorem would lead to the following: Any Gauss-Manin system of a polynomial of two variables can be made Fuchsian if one chooses a suitable basis in the cohomologies.

For the entire collection see [Zbl 0940.00039].

The theorem would lead to the following: Any Gauss-Manin system of a polynomial of two variables can be made Fuchsian if one chooses a suitable basis in the cohomologies.

For the entire collection see [Zbl 0940.00039].

Reviewer: Masaaki Yoshida (Fukuoka)

### MSC:

34M15 | Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain |

34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |

34M25 | Formal solutions and transform techniques for ordinary differential equations in the complex domain |