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We show that the Waring rank of the 3 × 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 symmetric cactus rank of the 3 × 3 permanent is at least 14.

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This is an author-produced, peer-reviewed version of this article. © 2020, Elsevier. Licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 International license. The final, definitive version of this document can be found online at Linear Algebra and Its Applications, doi: 10.1016/j.laa.2019.11.007

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