FactorLibrary: From Polynomials to Circuits via Recursive Subgoals

A new reinforcement learning system called FactorLibrary can find minimal arithmetic circuits for polynomials over finite fields, a problem central to algebraic complexity theory, according to a preprint posted to arXiv on June 24, 2026 [1][2]. The work, authored by Michael Ruofan Zeng, reframes the search for minimal arithmetic circuits as a reinforcement learning problem that can be approached from two directions: bottom-up and top-down [1][2]. The central challenge is a combinatorial search space that expands rapidly as problem size grows [2]. To manage this, the system introduces FactorLibrary, a mechanism that stores factorizable subexpressions and reuses them as subgoals across training episodes [1][2]. Three agent configurations were tested. A bottom-up agent used Gumbel-PPO-MCTS, while two top-down agents employed PPO+MCTS and SAC, respectively [2]. The PPO+MCTS top-down agent delivered the most stable results, finding certified optimal circuits up to complexity 8 with a success rate of 91.8% [1][2]. The preprint, which runs to 2,701 KB, was submitted to arXiv on June 24, 2026 [1]. arXiv, founded in 1991, is an open-access repository that hosts electronic preprints across mathematics, physics, computer science, and other fields without peer review [6]. As of November 2024, the platform was receiving roughly 24,000 new articles per month [6]. The repository surpassed two million total articles by the end of 2021 [6]. The FactorLibrary paper appears alongside a suite of experimental community tools offered through arXivLabs, a framework that allows third-party collaborators to build features directly on the arXiv abstract page [4][5]. These tools include the Bibliographic Explorer, which maps citation trees, and the CORE Recommender, which surfaces related open-access papers from a global network of repositories [4][5]. arXivLabs operates under guidelines that require partners to share arXiv’s values of openness, community, excellence, and user data privacy, and collaborators receive only minimal, anonymized user data [4].