Learning to Coordinate Symbolic Tools: LLM Agents for Verified Sum-of-Squares Certificates

We study how language-model agents coordinate symbolic tools to construct weighted sum-of-squares certificates for polynomial nonnegativity. The agent combines supervised algebraic training, reinforcement learning with symbolic rewards, and SymPy tools for polynomial transformations. Every proposed certificate is verified exactly by expansion and coefficient comparison. On held-out synthetic problems from the same generator, the trained agent with tools achieves 78.96% verified success on weighted sum-of-squares tasks, compared with 44.73% for the base model using the same tools. The study illustrates how domain training, executable tools, and verifier feedback can support mathematical reasoning with checkable outputs.

arXiv, arXiv:2608.00326