A. Li and S. Saydam (USA)

Time Series, Polynomial Models, Gr¨obner Basis

Consider a time series T and the set of polynomial models of T. We discuss two types of linearalities of T. The ﬁrst type is measured by the maximal number of linear mem bers of a polynomialmodel may have, denoted LN(T). An upper bound for LN(T) is given. The other is measured by the number of linear elements in a Gr¨obner Basis G of the ideal vanishing at all points of T, denoted LIN(G). Note that for each selected term order on the monomials of F[x1, . . ., xn], there is a unique generating set, called the reduced Gr¨obner Basis, for the vanishing ideal mentioned above. We give a method to ﬁnd linear members in G with respect to any term order. When selecting a graded term order (total degree prefered), we give a formula for the car dinality of LIN(G). Sample models are illustrated to sup port the theorems and propositionsand they are constructed using the Buchberger M¨oller Algorithm.

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