Improved Debordering of Waring Rank

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