Effective Strassmann Certificates for Local $p$-adic Dynamical Mordell--Lang Interpolants

Abstract

The $p$-adic method for Dynamical Mordell--Lang often reduces a residue class of an orbit to the zero set of a locally analytic interpolating function. This paper assumes the standard interpolation, Strassmann, Mahler, and Weierstrass tools, and studies the effective local zero-bound problem that remains after interpolation: certifying the Strassmann index of the resulting one-variable analytic function. We give finite certificates for this index, including finite-data, finite-precision, refined-tail, adaptive residue-class, one-shot, and first-order escape criteria. Since the Strassmann index is a rigorous upper bound for zeros in $\Zp$ and, through Weierstrass preparation, a root count on the closed disc over $\Cp$, these certificates give checkable stopping criteria for local orbit-intersection computations. A residue-class zooming principle replaces a congruence class of times by the iterate $f^{p^h}$, gaining $h$ additional powers of $p$ in the certificate tails. We also introduce an arc-ideal viewpoint for target varieties defined by several equations, replacing a chosen hypersurface bound by the one-variable gcd of all defining equations along the interpolated orbit. In dimension one, the method identifies the certified Strassmann index with the corresponding local Weierstrass root count in the orbit ball. Applications include certified bounds for intersections of non-fixed power-map orbits with finite target sets, and root-of-unity avoidance for maps tangent to the identity at torsion units.