We present realization theory for a class of autonomous discrete-time hybrid systems called semi-algebraic hybrid systems. These are systems in which the state and output equations associated with each discrete state are defined by polynomial equalities and inequalities. We first show that these systems generate the same output as semi-algebraic systems and implicit polynomial systems. We then derive necessary and almost sufficient conditions for existence of an implicit polynomial system realizing a given time-series data. We also provide a characterization of the dimension of a minimal realization as well as an algorithm for computing a realization from a given time-series data.
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