# Discovering Hidden Algebraic Structures via Transformers with Rank-Aware Beam GRPO

arXiv: https://arxiv.org/abs/2508.15766 · HTML: https://mlsanity.com/paper/2508.15766/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Mathematical reasoning, one of 17 directions in LLM reasoning.
- Builds on 16 of its 38 references that are in the map.
- First posted on arXiv 21 Aug 2025.

## Placed in

- LLM reasoning / [Mathematical reasoning](https://mlsanity.com/lr/mathematical-reasoning/)

## Builds on (most cited first, 16 of 16)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Sequence Level Training with Recurrent Neural Networks](https://mlsanity.com/paper/1511.06732/) (2015)
- [Generative Language Modeling for Automated Theorem Proving](https://mlsanity.com/paper/2009.03393/) (2020)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Beam Search Strategies for Neural Machine Translation](https://mlsanity.com/paper/1702.01806/) (2017)
- [A Neural Network Solves, Explains, and Generates University Math Problems by Program Synthesis and Few-Shot Learning at Human Level](https://mlsanity.com/paper/2112.15594/) (2021)
- [When to Finish? Optimal Beam Search for Neural Text Generation (modulo beam size)](https://mlsanity.com/paper/1809.00069/) (2018)
- [IK-Geo: Unified Robot Inverse Kinematics Using Subproblem Decomposition](https://mlsanity.com/paper/2211.05737/) (2022)

## Built on by (most cited first, 1 of 1)

- [FutureWorlds: Learning Robotic World Models from Alternative Futures](https://mlsanity.com/paper/2610.01019/) (2026)

## Abstract (arXiv)

Recent efforts have extended the capabilities of transformers in logical reasoning and symbolic computations. In this work, we investigate their capacity for non-linear latent pattern discovery in the context of functional decomposition, focusing on the challenging algebraic task of multivariate polynomial decomposition. This problem, with widespread applications in science and engineering, is proved to be NP-hard, and demands both precision and insight. Our contributions are threefold: First, we develop a synthetic data generation pipeline providing fine-grained control over problem complexity. Second, we train transformer models via supervised learning and evaluate them across four key dimensions involving scaling behavior and generalizability. Third, we propose Beam Grouped Relative Policy Optimization (BGRPO), a rank-aware reinforcement learning method suitable for hard algebraic problems. Finetuning with BGRPO improves accuracy while reducing beam width by up to half, resulting in approximately 75% lower inference compute. Additionally, our model demonstrates competitive performance in polynomial simplification, outperforming Mathematica in various cases.