Discrete Variational Problems: A Differential-Algebraic Framework and Constructive Solutions

Key Points

• To establish a differential-algebraic framework for discrete variational problems on graphs and discrete structures.
• Extended Ritt-Kolchin differential algebra theory to discrete environments.
• Developed constructive representation theorems for discrete variational problems.
• Implemented computational algorithms with interval arithmetic verification protocols.
• Demonstrated a unified constructive representation theorem applicable to various discrete variational problems.
• Presented explicit solution formulas and verified higher-order variations through spectral theory and correction terms.