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| Format: | Recurso digital |
| Language: | English |
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2026
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| Online Access: | https://doi.org/10.5281/zenodo.18136368 |
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| _version_ | 1866901384392081408 |
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| author | liu, shifa |
| author_facet | liu, shifa |
| contents | <p>In response to the celebrated negative resolution of Hilbert’s Tenth Problem (H10) over the ring of integers Z (Matiyasevich’s Theorem), this paper establishes a novel, constructive, and affirmative answer by transferring the problem into richer extended algebraic/analytic structures. We systematically develop two parallel theoretical frameworks: a model-theoretic differential-algebraic framework and an arithmetic-geometric computable-analysis framework. In the first framework, based on the theory of differentially closed fields (ACFA0), we prove that the problem of deciding the existence of solutions for an arbitrary Diophantine equation becomes theoretically decidable. Moreover, utilizing differential Galois theory and representation theory, we construct a precise, explicit analytic parameterization of the solution set (denoted Path B). In the second framework, based on a height-bounded arithmetic differential closure, we likewise establish theoretical decidability and provide three complementary families of explicit analytic parameterizations: one based on arithmetic derivatives and p-adicanalysis (Path A), a fully computable parameterization using modular filtering (Path C), and an optimized parameterization combining height-aware encoding with rapid convergence (Path D). We prove rigorous transformation theorems connecting these four paths, revealing that they belong to a unified representation theory: the solution set (or its height-bounded subset) can be globally approximated or exactly represented by a family of computable analytic functions on a finite-dimensional complex parameter space. This work not only provides a new “relativized” affirmative answer to H10 but also builds a robust bridge connecting discrete number theory, continuous analysis, model theory, and computable mathematics, opening a new paradigm for Diophantine analysis centered on explicitness, computability, and structural understanding.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18136368 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Affirmative Resolution of Hilbert's Tenth Problem in Extended Algebraic Structures: Decidability and Construction of Explicit Analytic Solutions liu, shifa Hilbert's Tenth Problem; Decidability; Differentially Closed Fields; Arithmetic Deriva tives; Explicit Analytic Solutions; Model Theory; Computable Analysis; Height Functions; Unified Frame work <p>In response to the celebrated negative resolution of Hilbert’s Tenth Problem (H10) over the ring of integers Z (Matiyasevich’s Theorem), this paper establishes a novel, constructive, and affirmative answer by transferring the problem into richer extended algebraic/analytic structures. We systematically develop two parallel theoretical frameworks: a model-theoretic differential-algebraic framework and an arithmetic-geometric computable-analysis framework. In the first framework, based on the theory of differentially closed fields (ACFA0), we prove that the problem of deciding the existence of solutions for an arbitrary Diophantine equation becomes theoretically decidable. Moreover, utilizing differential Galois theory and representation theory, we construct a precise, explicit analytic parameterization of the solution set (denoted Path B). In the second framework, based on a height-bounded arithmetic differential closure, we likewise establish theoretical decidability and provide three complementary families of explicit analytic parameterizations: one based on arithmetic derivatives and p-adicanalysis (Path A), a fully computable parameterization using modular filtering (Path C), and an optimized parameterization combining height-aware encoding with rapid convergence (Path D). We prove rigorous transformation theorems connecting these four paths, revealing that they belong to a unified representation theory: the solution set (or its height-bounded subset) can be globally approximated or exactly represented by a family of computable analytic functions on a finite-dimensional complex parameter space. This work not only provides a new “relativized” affirmative answer to H10 but also builds a robust bridge connecting discrete number theory, continuous analysis, model theory, and computable mathematics, opening a new paradigm for Diophantine analysis centered on explicitness, computability, and structural understanding.</p> |
| title | Affirmative Resolution of Hilbert's Tenth Problem in Extended Algebraic Structures: Decidability and Construction of Explicit Analytic Solutions |
| topic | Hilbert's Tenth Problem; Decidability; Differentially Closed Fields; Arithmetic Deriva tives; Explicit Analytic Solutions; Model Theory; Computable Analysis; Height Functions; Unified Frame work |
| url | https://doi.org/10.5281/zenodo.18136368 |