SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers

Kechen Li; Wenqi Zhu; Coralia Cartis; Tianbo Ji; Shiwei Liu

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arXiv · 2502.20545

SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers

Kechen Li·Wenqi Zhu·Coralia Cartis·Tianbo Ji·Shiwei Liu

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Abstract

Large Language Models (LLMs) have achieved human-level proficiency across diverse tasks, but their ability to perform rigorous mathematical problem solving remains an open challenge. In this work, we investigate a fundamental yet computationally intractable problem: determining whether a given multivariate polynomial is nonnegative. This problem, closely related to Hilbert's Seventeenth Problem, plays a crucial role in global polynomial optimization and has applications in various fields. First, we introduce SoS-1K, a meticulously curated dataset of approximately 1,000 polynomials, along with expert-designed reasoning instructions based on five progressively challenging criteria. Evaluating multiple state-of-the-art LLMs, we find that without structured guidance, all models perform only slightly above the random guess baseline 50%. However, high-quality reasoning instructions significantly improve accuracy, boosting performance up to 81%. Furthermore, our 7B model, SoS-7B, fine-tuned on SoS-1K for just 4 hours, outperforms the 671B DeepSeek-V3 and GPT-4o-mini in accuracy while only requiring 1.8% and 5% of the computation time needed for letters, respectively. Our findings highlight the potential of LLMs to push the boundaries of mathematical reasoning and tackle NP-hard problems.

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Kechen Li, Wenqi Zhu, Coralia Cartis, Tianbo Ji, Shiwei Liu. 2025-02-27. SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers. https://arxiv.org/abs/2502.20545

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