# Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley

## TL;DR

This paper investigates the limitations of the shifted partial derivatives method in algebraic complexity, proving it cannot separate the permanent from the determinant even without padding, and introduces explicit polynomials with maximal partial derivatives.

## Contribution

It extends the no-go results to unpadded models and provides explicit polynomial examples with maximal partial derivatives and higher symmetric border rank lower bounds.

## Key findings

- Shifted partial derivatives method cannot separate permanent from determinant without padding.
- Constructed explicit polynomials with maximal space of partial derivatives.
- Applied Koszul flattenings to obtain higher symmetric border rank lower bounds.

## Abstract

We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman] it was shown that the method of shifted partial derivatives cannot be used to separate the padded permanent from the determinant. Mulmuley asked if this "no-go" result could be extended to a model without padding. We prove this is indeed the case using the iterated matrix multiplication polynomial. We also provide several examples of polynomials with maximal space of partial derivatives, including the complete symmetric polynomials. We apply Koszul flattenings to these polynomials to have the first explicit sequence of polynomials with symmetric border rank lower bounds higher than the bounds attainable via partial derivatives.