行列式复杂度的二次下界
A Quadratic Lower Bound on Determinantal Complexity
- Tata Institute of Fundamental Research（塔塔基础研究所）
- Efi Arazi School of Computer Science, Reichman University（雷希曼大学埃菲·阿拉齐计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对幂和多项式给出其行列式复杂度的二次下界，提供简短自包含的证明，并指出先前AI辅助证明难以验证。
AI中文摘要：
我们证明了在复数域上，幂和多项式 $\sum_{i=1}^n x_i^n$ 的行列式复杂度具有 $\Omega(n^2)$ 的下界。类似的结果在 Sheshadri 最近的一篇论文（arXiv:2606.13628）中被声称，该论文采用了人工智能辅助且由人工智能撰写的证明。假设该证明的正确性，这将是这一基础代数问题中任意显式多项式的首个超线性下界。然而，尽管付出了相当大的努力，本笔记的作者仍无法理解并验证 arXiv:2606.13628 中的论证细节。我们在此提供的证明简短、（几乎）自包含且看似更简单。
英文摘要：
We prove an $Ω(n^2)$ lower bound on the determinantal complexity of the power sum polynomial $\sum_{i=1}^n x_i^n$ over the field of complex numbers. A similar result was claimed in a recent paper of Sheshadri (arXiv:2606.13628), via an AI-assisted and AI-written proof. Assuming its correctness, this was the first super-linear lower bound for this fundamental algebraic problem for any explicit polynomial. However, the authors of this note were unable to follow the details and verify the argument in arXiv:2606.13628, in spite of considerable effort on their part. The proof we provide here is short, (almost) self-contained and seemingly simpler.
