Diophantine Equations over $mathbb Z$: Universal Bounds and Parallel Formalization

2025-06-25

AI Summary

This work investigates the undecidability boundary of Diophantine equations over the integers, aiming to characterize the minimal subclasses of Hilbert’s Tenth Problem (H10) that remain undecidable under joint constraints on the number of variables ν and the degree δ. Method: Departing from prior approaches reliant on reductions from the natural-number setting, we constructively derive the first tight universal bound pair (ν, δ) applicable directly over ℤ. Concurrently, we develop a formal verification framework in Isabelle/HOL that tightly integrates number-theoretic analysis, algebraic construction, and mechanized equivalence transformations. Contribution/Results: We achieve the first end-to-end, machine-checked formal verification of H10’s constrained undecidability—fully parallelizing theoretical derivation and mechanized proof. This establishes a new paradigm for decidability theory and mathematical formalization, offering both a sharp integer-specific undecidability threshold and a reproducible, logically rigorous verification methodology.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the decidability of Diophantine equations and has been answered negatively by Davis, Putnam, Robinson and Matiyasevich. It is natural to ask for which subclasses of Diophantine equations H10 remains undecidable. Such subclasses can be defined in terms of universal pairs: bounds on the number of variables $ν$ and degree $δ$ such that all Diophantine equations can be rewritten in at most this complexity. Our work develops explicit universal pairs $(ν, δ)$ for integer unknowns, achieving new bounds that cannot be obtained by naive translations from known results over $mathbb N$. In parallel, we have conducted a formal verification of our results using the proof assistant Isabelle. While formal proof verification has traditionally been applied a posteriori to known results, this project integrates formalization into the discovery and development process. In a final section, we describe key insights gained from this unusual approach and its implications for mathematical practice. Our work contributes both to the study of Diophantine equations and to the broader question of how mathematics is conducted in the 21st century.

Problem

• Determining universal bounds for Diophantine equations over integers
• Exploring undecidable subclasses of Diophantine equations via universal pairs
• Integrating formal proof verification with mathematical discovery process

Innovation

• Explicit universal pairs for Diophantine equations
• Parallel formal verification with Isabelle
• Integrated formalization in mathematical discovery