Differential Algebraic Framework for Cyclic Algebras: A Constructive Approach to Cyclic Differential Equations
shifa liu · Zenodo (CERN European Organization for Nuclear Research) · 2025
This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to differential equations on cyclic algebras. Building on the theory of differential algebraic closures for geometric PDEs and exterior differential equations, we develop a systematic approach for cyclic algebras that incorporates their distinctive algebraic and combinatorial structures.We define the cyclic differential closure KCyc through a recursive adjunction process that incorporates cyclic eigenvectors, cyclic fundamental solutions, and combinatorial correction terms derived from cyclic symmetry. The construction provides explicit solution representations for cyclic differential equations with certified convergence in appropriate cyclic Sobolev norms.Our main contributions include: (1) a constructive definition of the cyclic differential closure with complete existence and uniqueness proofs, fundamentally extending classical differential closure theories of Kolchin and Pommaret by incorporating cyclic algebraic structures; (2) explicit solution formulas combining particular solutions and cyclic eigenbasis expansions with certified convergence rates;(3) combinatorial correction coefficients ensuring exact cancellation of cross terms while preserving cyclic symmetry, derived from cyclic representation theory and q-calculus; (4) efficient O(nlogn) computational algorithms with certified error bounds using cyclic interval arithmetic; and (5) extensions to cyclic cohomology,noncommutative geometry, and quantum cyclic algebras with explicit representatives for topological invariants.The framework bridges differential algebra, representation theory, and noncommutative geometry while maintaining mathematical rigor and computational efficiency, providing both theoretical completeness and practical computability for cyclic differential equations.