The Minimum Polynomial of an Algebraic Solution of Abel's problem

Abel's problem is the question how to solve the equation y'=a*y where a is an algebraic function in x. Suppose that the solution y=exp(int(a,x)) is an algebraic function as well. Then the minimum polynomial for y could in principle be obtained by integrating the algebraic function a, which is a problem for which algorithms exist. However, the algorithms for integration of algebraic functions can take a very long time. This paper shows how to resolve this problem. The idea is to use evala(Trace(RootOf(...)^i)). Note that this idea has other applications as well, such as factoring polynomials.

The algorithm given in this paper is useful for calculating algebraic solutions of linear differentials equations. Note that this could be useful for explicitly constructing polynomials in the inverse Galois problem, because for finite groups there exists a good explicit method for the inverse Galois problem for linear differential equations by van der Put and Ulmer. Conversely, when the polynomial is constructed we can also verify that it has the correct Galois group over C(x) by computing the monodromy, for which an implementation is available as well.

An illustration in Maple V.5 for Galois group H72.

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