A bound for the Waring rank of the determinant via syzygies

Mats Boij, Zach Teitler

Published 2019-08-23Version 1

We show that the Waring rank of the $3 \times 3$ determinant, previously known to be between $14$ and $20$, 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 $14$.