A Lower Bound on Determinantal Complexity

TL;DR

This paper establishes a new lower bound on the determinantal complexity of a specific polynomial, showing it is at least 1.5 times the number of variables minus 3, advancing understanding of polynomial complexity.

Contribution

It provides the first super linear lower bound for the determinantal complexity of an explicit polynomial, surpassing previous bounds of n+1.

Findings

Lower bound of 1.5n - 3 for the polynomial sum of x_i^n

Improves upon the previous bound of n+1

Advances the understanding of polynomial determinantal complexity

Abstract

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P = Det(M)$. We prove that the determinantal complexity of the polynomial $\sum_{i = 1}^n x_i^n$ is at least $1.5n - 3$. For every $n$-variate polynomial of degree $d$, the determinantal complexity is trivially at least $d$, and it is a long standing open problem to prove a lower bound which is super linear in $\max\{n,d\}$. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than $\max\{n,d\}$, and improves upon the prior best bound of $n + 1$, proved by Alper, Bogart and Velasco [ABV17] for the same polynomial.