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This paper studies polynomial models of time series. The focus will be on minimum degrees of polynomial modeling, in particular, the minimum degrees for arbitrary tail row. The paper proves decomposition theorems to reduce the associated matrices of time series to various matrix blocks. It introduces an augmented matrix of the associated matrix and gives a simple equivalent condition for existence of linear models. Moreover, it provides a new algorithm to get polynomial models, which improves the upper bound on the minimum degrees to $\le m-\bar l+1$ for an $m+1$ step time series with its augmented matrix of rank $\bar l$.

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