When working on some problem, I have end up with a following situation. Suppose $P(z)=z^d+a_{d-1}z^{d-1}+\ldots+a_1z+a_0$ is a complex polynomial $d\geq2$ and that $\gamma$ and $\delta$ are non-zero complex numbers and that $\gamma^2+\delta\neq0$ . Let us define $H:\mathbb{C}^2\rightarrow\mathbb{C}^2$ by $H(z,w)=(P(z)-\delta w,z)$ (Henon map). For easier formulation let us write $(z_n,w_n)=H(z_{n-1},w_{n-1})$ where $z_0=z$ and $w_0=w$. Obviously $z_{n-1}$ is a function of $z$ and $w$. ( we use the same notation for $(\zeta_n,\eta_n)=H(\zeta_{n-1},\eta_{n-1})$ )
Now I end up with a sequence of complex polynomials in 4 variables:
$Q_0(z,w,\zeta,\eta)=\gamma(w-\eta)-(z-\zeta)$
and for $n\geq1$
$Q_n(z,w,\zeta,\eta)=\gamma^{2-n}(P(z_{n-1})-P(\zeta_{n-1}))-(\gamma^2+\delta)(z-\zeta)$
If I define ideals $I_k=<Q_0,\ldots,Q_k>$ (generated by this polynomials), then we know that the chain of this ideals $I_0<I_1<I_2\ldots$ eventually stops at some number $k=N$. I know that in general this $N$ is not possible to determine, but I would belive that in this nice setting one should be able to find an upper bound. If we set $U=\{(z,w,z,w)|z,w\in\mathbb{C} \}$, and if $V$ denotes an algebraic set defined by $I_N$ (where chain stops) then results show that $\overline{V\backslash U}$ is 0 dimesional algebraic set (hence a finite set of points).
So my question would be, do you have any suggestions how to tackle this problem (how to find an upper bound)?
One could start computig by hand for $P(z)=z^2+c$ and get a result or by computing Grobner basis, but this gets quite messy when one starts to play with higher degree polynomials. So it woud be nice to have some theoretical tools to work with.
Thank you!
