# Computing Galois Groups of Completely Reducible Differential Equations

@article{Compoint1999ComputingGG, title={Computing Galois Groups of Completely Reducible Differential Equations}, author={Elie Compoint and Michael F. Singer}, journal={J. Symb. Comput.}, year={1999}, volume={28}, pages={473-494} }

We give an algorithm to calculate a presentation of the Picard?Vessiot extension associated to a completely reducible linear differential equation (i.e. an equation whose Galois group is reductive). Using this, we show how to compute the Galois group of such an equation as well as properties of the Galois groups of general equations.

## 41 Citations

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## References

SHOWING 1-10 OF 65 REFERENCES

Lectures on differential Galois theory

- Mathematics
- 1994

Differential ideals The Wronskian Picard-Vessiot extensions Automorphisms of Picard-Vessiot extensions The structure of Picard-Vessiot extensions The Galois correspondence and its consequences The…

Differential Equations and Algebraic Relations

- MathematicsJ. Symb. Comput.
- 1998

An algorithm is given for the construction of a system of generators of the ideal of algebraic relations, with coefficients inC(z), among the entries of a fundamental matrix of solutions of Ly=0, starting from the data of aC-algebra basis of theGinvariant polynomials with coefficient inCinnvector variables.

Linear Algebraic Groups

- Mathematics
- 1975

Algebraic geometry affine algebraic groups lie algebras homogeneous spaces chracteristic 0 theory semisimple and unipoten elements solvable groups Borel subgroups centralizers of Tori structure of…

An algorithm for computing invariants of differential Galois groups

- Mathematics, Computer Science
- 1997

Direct methods for primary decomposition

- Mathematics
- 1992

SummaryLetI be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical ofI, for localizingI with respect to another ideal, and…

Factorization of Differential Operators with Rational Functions Coefficients

- MathematicsJ. Symb. Comput.
- 1997

This method solves the main problem in Beke's factorization method, which is the use of splitting fields and/or Grobner basis.

Gröbner Bases and Primary Decomposition of Polynomial Ideals

- Mathematics, Computer ScienceJ. Symb. Comput.
- 1988

Rational Solutions of the Mixed Differential Equation and Its Application to Factorization of Differential Operators

- MathematicsISSAC
- 1996

A fast method to compute the rational solutions of a certain differential equation that will be called the mixed differential equation can be applied to speed up the factorization of completely reducible linear differential operators with rational functions coefficients.

Algorithmic algebraic number theory

- MathematicsEncyclopedia of mathematics and its applications
- 1989

This chapter discusses the embedding of commutative orders into the maximal order of constructive algebraic number theory, and some of the methods used to derive these orders.