Linearality of polynomial models of discrete time series

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 members of a polynomial model 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 Gröbner 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, ..., x_n], there is a unique generating set, called the reduced Gröbner 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 cardinality of LIN(G). Sample models are illustrated to support the theorems and propositions and they are constructed using the Buchberger Möller Algorithm.