Symbolic-numeric algorithms in differential Galois theory (Joris van der Hoeven)

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Classically, computations in differential Galois theory are mainly done using algebraic 
techniques. An interesting density theorem by Ramis-Martinet and Schlesinger, based
 on previous work by Schlesinger and Ecalle, provides the differential Galois group as
 the closure of a set of generalized monodromy matrices. The entries of such matrices
are transcendental holonomic constants, which can be approximated rapidly up to any
 precision. This provides an effective means to translate questions about the
 differential Galois groups into linear algebra or questions about linear algebraic
 groups. One difficulty, however, is that no zero-test is available for the constants we
 are working with. Nevertheless, in our talk, we will present an algorithm for the
 factorization of linear differential operators which avoids this difficulty, 
based on the fact that we can test whether a candidate factorization 
is a genuine one.