Unique Extrapolation of Polynomial Recurrences

Let a sequence of k-dimensional vectors ${\bf x}_0 ,{\bf x}_1 , \cdots $ (over a ring A) be determined by a polynomial recurrence of form ${\bf x}_n = T({\bf x}_{n - 1} )$, where $T:A^k \to A^k $ itself is known to be a polynomial map in k variables of degree at most d but is otherwise unknown. We show that there is a finite N such that the entire sequence $\{ {\bf x}_n :n \geqq 0\} $ can be deduced from the first $N + 1$ terms ${\bf x}_0 ,{\bf x}_1 , \cdots ,{\bf x}_N $ alone. The number $N = \phi (d,k,A)$ depends on d and k and the ring A but not on T.Let $\phi ^ * (d,k)$ denote the maximum of $\phi (d,k,A)$ over all commutative rings with unit. Then we show that $\phi ^ * (d,k) < \infty $. In particular, $\phi ^ * (d,1) = d + 1$ and $\phi ^ * (1,k) = k + 1$. In the general case $\phi ^ * (d,k) \geqq \left( {\begin{array}{*{20}c} {k + d} \\ k \\ \end{array} } \right)$ and equality does not always hold because $\phi ^ * (2,2) \geqq 7$. In addition, we show that for each k that $\max \{ \phi (d,k,{\bf F})...