Abstract
This paper studies rational and Liouvillian first integrals of homogeneous linear differential systems Y′=AY over a differential field k. Following [26], our strategy to compute them is to compute the Darboux polynomials associated with the system. We show how to explicitly interpret the coefficients of the Darboux polynomials as functions on the solutions of the system; this provides a correspondence between Darboux polynomials and semi-invariants of the differential Galois groups, which in turn gives indications regarding the possible degrees for Darboux polynomials (particularly in the completely reducible cases). The algorithm is implemented and we give some examples of computations.
Research supported by the École Polytechnique, the CNRS GDR 1026 (MEDICIS), the GDR-PRC 967 (Math-Info), and the CEC ESPRIT BRA contract 6846 (POSSO). The computations have been performed on the machines supplied by MEDICIS.
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Weil, JA. (1995). First integrals and Darboux polynomials of homogeneous linear differential systems. In: Cohen, G., Giusti, M., Mora, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1995. Lecture Notes in Computer Science, vol 948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60114-7_37
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DOI: https://doi.org/10.1007/3-540-60114-7_37
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