Abstract
In this paper a polynomial map from Cn to Cm is studied in order to investigate if it is injective out of a set of measure zero. We propose a procedure, based on truncated Gröbner basis computations, which when successful, allows to reduce the problem to an easier map, and so gives a speed-up of the general algorithms using Gröbner basis techniques. Moreover, for the special case of a polynomial map from Cn to Cn where the polynomials are at most quadratic, we propose two criteria for non-injectivity based on the structure of the Jacobian matrix and requiring only basic symbolic computations.
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References
Audoly S., D'Angiò L.: On the identifiability of linear compartmental systems: a revisited transfer function approach based on topological properties. Math. Biosci.66, 201–228 (1983)
Cobelli C., Lepschy A.: Romanin-Jacur G., Identifiability of experimental systems and related structural properties. Math. Biosci.44, 1–18 (1979)
D'Angiò L.: On some topological properties of a strongly connected compartmental system with application to the identifiability problem. Math. Biosci.76, 207–220 (1985)
Giovini A., Niesi G.: CoCoA: a user-friendly system for commutative algebra. Proc. DISCO '90. Lecture Notes in Computing Science, Vol. 429, pp. 20–29. Berlin, Heidelberg, New York: Springer 1990
Ollivier F.: Inversibility of rational mappings and structural identifiability in automatics. Proc. ISSAC 89, ACM (1989)
Ollivier F.: Le probléme de l'identifiabilité structurelle globale: étude theorique, méthodes effectives et bornes de complexité, Thése de Doctorat, Ecole Polytechnique (1990)
Raksany A., Lecourtier Y., Walter E., Venot A.: Identifiability and distinguishability testing via Computer Algebra. Math Biosci.77, 245–266 (1985).
Raksanyi A.: Utilisation du calcul formel pour l'étude des systèmes d'équations polynomiales (application en modélisation) Thèse de Doctorat, Université Paris-Dauphine (1986)
Shannon D., Sweedler M.: Using Gröbner bases to determine algebra membership, split surjective algebra homomorphisms and determine birational equivalence, J. Symb. Comp.6 (1988)
Shannon D., Sweedler M.: Using Gröbner bases to determine subalgebra membership, Preprint (1988)
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Audoly, S., Bellu, G., Buttu, A. et al. Procedures to investigate injectivity of polynomial maps and to compute the inverse. AAECC 2, 91–103 (1991). https://doi.org/10.1007/BF01810570
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DOI: https://doi.org/10.1007/BF01810570