Agentic Computational Algebra

We explore the use of AI agents for solving computational problems in algebra. For the problems we consider, the main challenge is not correctness but time: general-purpose algorithms provide rigorous guarantees but can be astronomically slow, and whether a problem is solvable in practice may depend on how a system is posed or which specialized tool is used. We study whether agents can automate these choices while learning from past verified attempts, and whether they can implement a fast reusable solver for known problem families. More broadly, computational algebra provides a useful testbed for agent tool use and memory, since it supplies infinitely many problems with verifiable answers and tunable difficulty. To this end, we describe concrete benchmark families of zero-dimensional polynomial systems with controlled shared structure, and we evaluate agents that can reformulate problems, invoke existing algebra tools, and retain experience across instances. Preliminary experiments show that agents can expand the frontier of solvable problems by discovering and reusing effective reformulations. However, model reasoning and memory maintenance can offset these gains, although compilation can amortize the cost of rediscovery across instances.