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

A bound for the Waring rank of the determinant via syzygies