Realizations of Polynomial Systems
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This paper is a continuation of the previous author s paper , where a new definition of a polynomial input-output system has been proposed. We have been interested in immersions of nonlinear systems into polynomial ones (immersions into simpler systems as affine or linear ones have been studied by Fliess and Kupka , Claude, Pliess and Isidori  and Claude . The observation algebra of a nonlinear system has appeared to be a useful tool for examination of this problem. Finite generatedness and regularity of the observation algebra are necessary and sufficient in order that the nonlinear system may be regularly immersed into an algebraically observable polynomial system. Here we study a more general case — without assumptions about regularity and observability. Now the observation algebra does not have to be finitely generated but has to be a subalgebra of a finitely generated algebra. However, we complete the result of  proving that if the observation algebra is finitely generated then there is an immersion into an algebraically observable polynomial system.
KeywordsNonlinear System Polynomial System Ring Homomorphism Polynomial Realization Minimal Realization
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