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Improved Debordering of Waring Rank

Amir Shpilka

TL;DR

This work tackles the gap between border Waring rank and Waring rank for homogeneous polynomials, proving that if $\underline{\mathrm{WR}}(f)=r$, then $\mathrm{WR}(f) \leq d \cdot r^{O(\sqrt{r})}$ (in fact shown as $\leq d \cdot r^{10\sqrt{r}}$ in the main argument). It introduces an $\varepsilon$-perturbed debordering technique and a diagonalization scheme that transform border decompositions into locally structured forms, enabling rank bounds via variable partitioning and derivatives. The main theorem improves the previous bound $\mathrm{WR}(f) \leq d \cdot 4^r$ from DuttaGIJL24 and sharpens our understanding of when border representations yield polynomially bounded Waring ranks. The results advance debordering theory for the symmetric tensor rank and have potential implications for computing ranks of degree-$d$ forms in algebraic complexity contexts.

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

Improved Debordering of Waring Rank

TL;DR

This work tackles the gap between border Waring rank and Waring rank for homogeneous polynomials, proving that if , then (in fact shown as in the main argument). It introduces an -perturbed debordering technique and a diagonalization scheme that transform border decompositions into locally structured forms, enabling rank bounds via variable partitioning and derivatives. The main theorem improves the previous bound from DuttaGIJL24 and sharpens our understanding of when border representations yield polynomially bounded Waring ranks. The results advance debordering theory for the symmetric tensor rank and have potential implications for computing ranks of degree- forms in algebraic complexity contexts.

Abstract

We prove that if a degree- homogeneous polynomial has border Waring rank , then its Waring rank is bounded by This result significantly improves upon the recent bound established in [Dutta, Gesmundo, Ikenmeyer, Jindal, and Lysikov, STACS 2024], which itself was an improvement over the earlier bound .

Paper Structure

This paper contains 6 sections, 8 theorems, 42 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Facts from DuttaGIJL24
  4. Improved debordering
  5. The case $d \geq r-1$.
  6. The case $d < r-1$.

Key Result

Theorem 1.1

Let $f \in \mathbb{C}[\mathbold{x}]_d$ be a homogeneous polynomial. If $\operatorname{\underline{\mathrm{WR}}}\left(f\right) = r$, then

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 2.1: Waring Rank
  • Definition 2.2: Border Waring Rank
  • Definition 2.3: Border Waring Rank Decomposition
  • Definition 2.4: Local Decomposition DuttaGIJL24
  • Lemma 2.5: Lemma 4 of DuttaGIJL24
  • Lemma 2.6: Lemma 6 of DuttaGIJL24
  • Lemma 2.7: Lemma 7 of DuttaGIJL24
  • Lemma 2.8: Lemma 10 of DuttaGIJL24
  • Lemma 2.9: Perturbing a Variable by $\varepsilon$
  • ...and 8 more