A bound for the Waring rank of the determinant via syzygies

TL;DR

This paper establishes a new lower bound for the Waring rank of the 3x3 determinant using syzygies of the apolar ideal, advancing understanding of polynomial decompositions.

Contribution

It introduces a novel method employing syzygies of the apolar ideal to improve lower bounds on the Waring rank of the determinant.

Findings

Waring rank of 3x3 determinant is at least 15

Cactus rank of 3x3 permanent is at least 14

New technique using syzygies of the apolar ideal

Abstract

We show that the Waring rank of the $3 \times 3$ determinant, previously known to be between $14$ and $18$, is at least $15$. We use syzygies of the apolar ideal, which have not been used in this way before. Additionally, we show that the cactus rank of the $3 \times 3$ permanent is at least $14$.