Unified Solving of Structured Diophantine Equations via Hierarchical Arithmetic Differential Closure: A Constructive and Effective Framework

This paper introduces a constructive framework for solving structured Diophantine equations using a novel hierarchical arithmetic differential closure (ADC). Building upon Buium’s arithmetic differential algebra, we develop a recursive con struction that incorporates p-adic derivative operators and arithmetic invariants under explicit height control. For a system of m Diophantine equations f(x) = 0 in n variables with maximal total degree d and a Galois symmetry group G, we prove that all solutions within a height bound B reside in an explicitly constructed arithmetic differential field extension Karith(B) whose degree and height are effectively bounded. Our main representation theorem expresses solutions through arithmetic critical value tensors and G-equivariant basis functions, with complete treatment of both non-singular and singular cases. The associated algorithms achieve quasi-polynomial complexity O((logB)O(1)) for systems with fixed dimension and treewidth, improving upon traditional methods for structured problems. We provide complete verification for Pell equations and singular systems, and outline extensions to more complex systems. The framework is shown to be consistent with the Matiyasevich-Davis-Putnam-Robinson theorem by restricting to equations with sufficient symmetry and explicit height bounds.