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March 7, 2026Open Access

A Comprehensive Generalization of the Fundamental Theorem of Algebra and Vieta's Theorem in the Framework of Difference Equations

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This monograph explores generalizations of algebraic theorems in difference equations, implying deep connections across mathematical fields.

The aim is to generalize the Fundamental Theorem of Algebra and Vieta's Theorem in the context of difference equations.
Explored linear constant-coefficient and variable-coefficient difference equations.
Established relations using Grassmann algebra and discrete Liouville formulas.
Developed $q$-deformed Liouville formulas and stochastic discrete Liouville formulas.
Proved that in constant-coefficient cases, discrete Liouville formulas are equivalent to Vieta's Theorem.
Developed a complete Galois-theoretic interpretation of Vieta's Theorem.
Showed connections between higher-order formulas and $ au$-functions in integrable systems.

shifa liu (2025) studied this question.

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This monograph explores generalizations of algebraic theorems in difference equations, implying deep connections across mathematical fields.