Inversibility of rational mappings and structural identifiability in automatics

We investigate different methods for testing whether a rational mapping f from kⁿ to k^m admits a rational inverse, or whether a polynomial mapping admits a polynomial one. We give a new solution, which seems much more efficient in practice than previously known ones using "tag" variables and standard basis, and a majoration for the degree of the standard basis calculations which is valid for both methods in the case of a polynomial map which is birational. We further show that a better bound can be given for our method, under some assumption on the form of f. Our method can also extend to check whether a given polynomial belongs to the subfield generated by a finite set of fractions. We then illustrate our algorithm, with a application to structural identifiability. The implementation has been done in the IBM computer algebra system Scratchpad II.