Explicit Hilbert's Nullstellensatz and Orbits in Polynomial Dynamics over Finite Fields

We use recent explicit versions of Hilbert's Nullstellensatz to answer several natural questions about reductions of orbits modulo a prime $p$ of polynomial dynamical systems defined over $\mathbb {Z}$. We first show that for sufficiently large primes $p$, the reduction modulo $p$ of a zero dimensional variety over $\mathbb Q$ remains zero dimensional over $\overline{\mathbb F}_p$.

We apply these estimates to studying cyclic points and, uder a certain natural condition on the intersections of such orbits over $\mathbb C$ corresponding to two distinct polynomial systems, we show that intersections modulo $p$ are rare. These results can be considered as modulo $p$ versions of recent results of D. Ghioca, T. J. Tucker, and M. E. Zieve in characteristic zero. However the underlying approach and techniques are different.

Joint work with Carlos D'Andrea, Igor E. Shparlinski and Martin Sombra.

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