Improved Debordering of Waring Rank

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 $\mathrm{WR}(f)\le d\cdot 4^r$ established in [Dutta, Gesmundo, Ikenmeyer, Jindal, and Lysikov, STACS 2024], which itself was an improvement over the earlier bound $\mathrm{WR}(f)\le d^r$.