The differential Hilbert function of a differential rational mapping can be computed in polynomial time

@inproceedings{Matera2002TheDH,
  title={The differential Hilbert function of a differential rational mapping can be computed in polynomial time},
  author={Guillermo Matera and Alexandre Sedoglavic},
  booktitle={ISSAC},
  year={2002}
}
We present a probabilistic seminumerical algorithm that computes the differential Hilbert function associated to a differential rational mapping. This algorithm explicitly determines the set of variables and derivatives which can be arbitrarily fixed in order to locally invert the differential mapping under consideration. The arithmetic complexity of this algorithm is polynomial in the input size. 

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