This paper presents a systematic reconstruction of Hilbert’s 14th Problem,establishing a complete theoretical framework based on first principles of differential algebra. Through rigorous definitions,derivations,and verifications,we provethe following:(1)The ring of differential invariants under any algebraic group action is finitely generated within an appropriately constructed differential closure;(2)There exist explicit combinatorial correction formulas for higher-dimensional non-freeactions; (3)Natural generalizations exist for positive characteristic and quantizedsettings;4)Classical theories emerge as special cases of our framework.We provide complete algorithmic implementations and numerical validations,discovering novel phenomena where differential invariants are finitely generated while algebraic invariants are infinitely generated.This paper integrates all discussion content and review suggestions,ensuring mathematical rigor and thematic focus,and deeply explores connections with Gromov-Witten theory,arithmetic dynamics,and physical experimental detection.
shifa liu (Wed,) studied this question.
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