Improved Debordering of Waring Rank

TL;DR

This paper establishes a new upper bound on the Waring rank of a homogeneous polynomial based on its border Waring rank, improving previous bounds significantly and advancing understanding of polynomial decomposition complexity.

Contribution

It proves a tighter upper bound on Waring rank in terms of border Waring rank, refining previous exponential bounds and contributing to polynomial decomposition theory.

Findings

Waring rank is bounded by d * r^{O(√r)} for polynomials with border Waring rank r.

This bound improves upon the previous exponential bounds in r.

The result advances the theoretical understanding of polynomial decomposition complexity.

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 ${\mathrm{WR}}({f}) \leq d^r$.