Improved Debordering of Waring Rank

Amir Shpilka

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arXiv · 2502.03150

Improved Debordering of Waring Rank

Amir Shpilka

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Abstract

We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $\underline{\mathrm{WR}}({f}) = r$, then its Waring rank is bounded by \[ {\mathrm{WR}}({f}) \leq d \cdot r^{O(\sqrt{r})}. \] This result significantly improves upon the recent bound ${\mathrm{WR}}({f}) \leq d \cdot 4^r$ established in [Dutta, Gesmundo, Ikenmeyer, Jindal, and Lysikov, STACS 2024], which itself was an improvement over the earlier bound ${\mathrm{WR}}({f}) \leq d^r$.

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Amir Shpilka. 2025-02-05. Improved Debordering of Waring Rank. https://arxiv.org/abs/2502.03150

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