AI 中文总结
研究动力 Mordell--Lang 的 \(p\)-adic 方法插值后的局部零界问题，给出确定单变量解析函数施特拉斯曼指数的有限证书及相关准则，还介绍弧理想观点，可用于轨道相交计算等，一维时能等同施特拉斯曼指数与局部魏尔斯特拉斯根数。
AI 中文摘要
动力 Mordell--Lang 的 \(p\)-adic 方法通常将轨道的一个剩余类简化为局部解析插值函数的零集。本文采用标准插值、施特拉斯曼、马勒和魏尔斯特拉斯工具，研究插值后剩余的有效局部零界问题：确定所得单变量解析函数的施特拉斯曼指数。我们给出了该指数的有限证书，包括有限数据、有限精度、精细尾部、自适应剩余类、一次性和一阶逃逸准则。这些证书为局部轨道相交计算提供了可检验的停止准则。还介绍了一种针对由多个方程定义的目标簇的弧理想观点。在一维情况下，该方法将认证的施特拉斯曼指数与轨道球中相应的局部魏尔斯特拉斯根数等同起来。应用包括非固定幂映射轨道与有限目标集相交的认证界，以及在挠单位处与恒等映射相切的映射的单位根避免。
英文摘要
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.
Comments28 pages
