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Consider a sequence of polynomials $Z_N(s,t,p)$ where $N$ is not necessarily the degree and hence not traditional (if its of any use $N$ is the degree only in the case of $N=1$). Suppose we are given a "linear" recurrence relation of the form $Z_N= \sum_{i=1}^{M} w_i(s,t,p)Z_{N-i}$ where $M$ is the order of the recurrence relation. We call this "linear" because we disallow terms such as $Z_{N-1}*Z_{N-2}$ or $Z_{N-3}^2$ and the coefficients are all polynomials.

In the problem I am studying I require a closed form solution for a particular non-traditional 3-variable polynomial recurrence relation of order 9 (i.e. $Z_N$ depends on the previous 9 terms). I'm certain research has been done on the general topic but have had a difficult time finding any. Perhaps I am using the wrong terminology? Any help on that would be appreciated.

Even better of course would be any direct information on these class of problems but of course I'll be happier with a reference as well.

Any info is welcome. Thanks.

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You have a linear recurrence over the field $\textbf{C}(s,t,p)$, where $s$, $t$, and $p$ are variables. There are standard methods for expressing the $N$'th term in terms of the roots of the associated polynomial $X^M-\sum w_i(s,t,p)X^{M-i}$. These roots will live in a finite extension field of $\textbf{C}(s,t,p)$. Geometrically, you'll be working in the function field of a 3-fold that is an $M$-fold cover of $\textbf{P}^3$. Then $Z_N$ is expressed in terms of power sums of the roots, which can, in principle, be expressed as a function in $\textbf{C}(s,t,p)$. The only slightly unusual feature is that you're working over a base field $\textbf{C}(s,t,p)$ that's a rational function field of three variables.

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Sounds like the approach is inherently one of algebraic geometry. Are there any standard ways of computing what the roots for polynomials in these fields would be? I would expect we would need a generalized notion of a radicals. And if an expression for the solutions don't always exist, would it be out of the question to look for an approximation to the roots which live in the closure of $Frac(Z[s,t,p])$ by using elements of the closure of $Frac(R[s,t,p])$? –  Christian Bueno Feb 16 '11 at 0:05
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