  In the paper "Uniformity of Mordell-Lang" by Vesselin Dimitrov, Philipp
Habegger and Ziyang Gao (arXiv:2001.10276), they use Silverman-Tate's Height
Inequality and they give a proof of the same which makes use of Cartier
divisors and hence drops the flatness assumption of structure morphisms of
compactified abelian schemes. However, their proof makes use of Hironaka's
theorem on resolution of singularities which is unknown for fields of positive
characteristic. We try to slightly modify their ideas, use blow-ups in place of
Hironaka's theorem to make the proof effective for any fields with product
formula where heights can be defined and any normal quasi-projective variety as
a base. We work in the more general set up of dynamical systems. As an
application we prove certain variant of Silverman's Specialization Theorem with
restricted hypotheses in higher dimensional bases.
