A Quadratic Lower Bound on Determinantal Complexity

2026-09-28

AI Summary

This study addresses the long-standing absence of concise and rigorous proofs for determinant complexity lower bounds of power sum polynomials, as well as the unverifiability of prior AI-assisted proof attempts. Drawing on algebraic complexity theory, this work presents a succinct, self-contained mathematical argument that refutes previous unreliable AI-generated proofs and rigorously establishes an Ω(n²) lower bound for the determinant complexity of power sum polynomials over the complex field. This contribution provides the first verifiable proof of a superlinear lower bound for an explicit polynomial, significantly simplifying the derivation while ensuring absolute reliability of the result. Ultimately, it sets a new paradigm for integrating formalized mathematical proofs with computational complexity theory.

Abstract

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.

Problem

• Determinantal Complexity
• Power Sum Polynomial
• Lower Bound
• Algebraic Complexity

Innovation

• Determinantal Complexity
• Power Sum Polynomial
• Quadratic Lower Bound
• Algebraic Complexity