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arXiv · 2503.06592

Automated Proof of Polynomial Inequalities via Reinforcement Learning

Banglong Liu·Niuniu Qi·Xia Zeng·Lydia Dehbi·Zhengfeng Yang
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Abstract

Polynomial inequality proving is fundamental to many mathematical disciplines and finds wide applications in diverse fields. Current traditional algebraic methods are based on searching for a polynomial positive definite representation over a set of basis. However, these methods are limited by truncation degree. To address this issue, this paper proposes an approach based on reinforcement learning to find a {Krivine-basis} representation for proving polynomial inequalities. Specifically, we formulate the inequality proving problem as a linear programming (LP) problem and encode it as a basis selection problem using reinforcement learning (RL), achieving a non-negative {Krivine basis}. Moreover, a fast multivariate polynomial multiplication method based on Fast Fourier Transform (FFT) is employed to enhance the efficiency of action space search. Furthermore, we have implemented a tool called {APPIRL} (Automated Proof of Polynomial Inequalities via Reinforcement Learning). Experimental evaluation on benchmark problems demonstrates the feasibility and effectiveness of our approach. In addition, {APPIRL} has been successfully applied to solve the maximum stable set problem.

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Banglong Liu, Niuniu Qi, Xia Zeng, Lydia Dehbi, Zhengfeng Yang. 2025-03-09. Automated Proof of Polynomial Inequalities via Reinforcement Learning. https://arxiv.org/abs/2503.06592

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