Differential Algebraic Framework for Morse Theory: Constructive Approaches to Critical Points and Gradient Flows
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2026
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| _version_ | 1866901239125508096 |
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| author | liu, shifa |
| author_facet | liu, shifa |
| contents | <p>This paper develops a comprehensive differential algebraic framework for Morse theory, extending the constructive methods of differential algebraic closures to the study of Morse functions, critical points, and gradient flows on smooth manifolds.We define the Morse differential closure KMorse, a differentially closed field extension that systematically incorporates Morse functions, their critical points, Hessian structures, gradient flow equations, and stable/unstable manifolds through a rigorous well-founded recursive construction.The construction employs the field of fractions of germs of smooth functions at critical points as the base structure for local computations, while for global computations on non-compact manifolds, we utilize function fields with appropriate growth conditions to ensure completeness. This approach avoids the limitations of meromorphic functions while preserving complete geometric information. We provide a countable construction that remains within ZFC set theory, avoiding reliance on uncountable ordinals. The framework handles non-degeneracy conditions through rigorous algebraic encoding techniques.Within this closure, we prove constructive existence theorems for local parameterizations of stable and unstable manifolds with complete convergence analysis,provide explicit formulas for Morse boundary operators with certified combinatorial counts, and establish computational methods for Morse theoretical invariants with rigorous error bounds. The framework bridges differential algebra, geometric analysis, and computational topology while maintaining complete mathematical rigor and providing explicit algorithmic implementations.All constructions are verified through detailed convergence proofs, combinatorial validations, and explicit examples with complete computations. The certification procedures employ interval arithmetic and validated numerical methods to ensure mathematical correctness. The framework is shown to be computationally feasible for moderate-dimensional manifolds through complexity analysis and optimization techniques.Connections to current research frontiers, including discrete Morse theory and persistent homology, are explicitly developed.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18126262 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Differential Algebraic Framework for Morse Theory: Constructive Approaches to Critical Points and Gradient Flows liu, shifa Morse theory, Differential algebraic closure, Critical points, Gra dient flows, Stable manifolds, Morse complex, Topological invariants, Constructive mathematics, Certified computation <p>This paper develops a comprehensive differential algebraic framework for Morse theory, extending the constructive methods of differential algebraic closures to the study of Morse functions, critical points, and gradient flows on smooth manifolds.We define the Morse differential closure KMorse, a differentially closed field extension that systematically incorporates Morse functions, their critical points, Hessian structures, gradient flow equations, and stable/unstable manifolds through a rigorous well-founded recursive construction.The construction employs the field of fractions of germs of smooth functions at critical points as the base structure for local computations, while for global computations on non-compact manifolds, we utilize function fields with appropriate growth conditions to ensure completeness. This approach avoids the limitations of meromorphic functions while preserving complete geometric information. We provide a countable construction that remains within ZFC set theory, avoiding reliance on uncountable ordinals. The framework handles non-degeneracy conditions through rigorous algebraic encoding techniques.Within this closure, we prove constructive existence theorems for local parameterizations of stable and unstable manifolds with complete convergence analysis,provide explicit formulas for Morse boundary operators with certified combinatorial counts, and establish computational methods for Morse theoretical invariants with rigorous error bounds. The framework bridges differential algebra, geometric analysis, and computational topology while maintaining complete mathematical rigor and providing explicit algorithmic implementations.All constructions are verified through detailed convergence proofs, combinatorial validations, and explicit examples with complete computations. The certification procedures employ interval arithmetic and validated numerical methods to ensure mathematical correctness. The framework is shown to be computationally feasible for moderate-dimensional manifolds through complexity analysis and optimization techniques.Connections to current research frontiers, including discrete Morse theory and persistent homology, are explicitly developed.</p> |
| title | Differential Algebraic Framework for Morse Theory: Constructive Approaches to Critical Points and Gradient Flows |
| topic | Morse theory, Differential algebraic closure, Critical points, Gra dient flows, Stable manifolds, Morse complex, Topological invariants, Constructive mathematics, Certified computation |
| url | https://doi.org/10.5281/zenodo.18126262 |