We prove an $\Omega(n^2)$ 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, 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 Sheshadri's proof in spite of considerable effort on their part.
The proof we provide here is short, (almost) self-contained and seemingly simpler.
\textbf{AI Use: }ChatGPT Astra was used by the authors on multiple occasions to parse through some of the parts of Sheshadri's proof in [Shesh2026] and the outline of the proof in this note came out of this exercise. The final exposition as well as some of the final details in this note are due to its human authors.
I'm Karthik Sheshadri, the author of [Shesh2026] (arXiv:2606.13628). I acknowledge feedback from the TCS community that my original article was difficult to follow. In response, I revised the article to include a much more detailed and hopefully more intuitive explanation, including worked examples and a guide for TCS readers. The article, that covers both the exact and border cases, is linked to here: https://doi.org/10.5281/zenodo.23196226
I am grateful to the authors of this note for their simpler exposition of the proof of the exact case. Their proof uses the same kernel lift at corank-one points, the same proxy system of kernel equations and bilinear derivative equations, the same removal of one redundant right-kernel equation, and the same multihomogeneous Bézout number, $\sum_k \binom{m}{k}\binom{m-1}{n-1-k}\binom{n-2}{k-1}$, as the determinant-side bound in [Shesh2026], and simplifies the step that controls the count of solutions, and may be useful for some TCS readers.
Interestingly, the substantive difference is the choice of system: setting $x_{n-1}=1$ makes every solution of the proxy system genuine, so the isolatedness and excess-intersection arguments of [Shesh2026] are not needed. The same count gives the constant $1/(4e)$.
As the authors state, ChatGPT Astra was used "to parse through some of the parts of Sheshadri's proof in [Shesh2026] and the outline of the proof in this note came out of this exercise." [Shesh2026] was posted in June 2026, and its symmetric version as ECCC TR26-097 on June 9, 2026.
