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Upper bounds for the rank of powers of quadrics

Cosimo Flavi

TL;DR

The paper establishes tight asymptotic control on the Waring rank of powers of quadrics $q_n^s$ in many variables, showing $ ext{rk}(q_n^s)=O(n^s)$ while the generic rank scales as $ ext{rk}_{ ext{gen}}(d)\sim n^{2s-1}$. It introduces a unifying framework based on partitions $oldsymbol{p}_k(s)$ and monomial symmetric polynomials $M_{oldsymbol{m}}$, enabling explicit decompositions and upper bounds, and proves that for all $s$ there is a threshold $n>(2s-1)^2$ beyond which the rank is subgeneric, i.e., $ ext{rk}(q_n^s)< rac{1}{n}inom{2s+n-1}{2s}$. The results hinge on catalecticant/apolarity techniques, symmetry-based decompositions, and asymptotic estimates for partition functions, providing both concrete decompositions for small $s$ and a general subgeneric-rank regime. These insights advance understanding of rank loci geometry and high-dimensional Waring problems in the context of quadratic powers.

Abstract

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any $s\in\mathbb{N}$, we prove that the $s$-th power of a quadratic form of rank $n$ grows as $n^s$. Furthermore, we demonstrate that its rank is subgeneric for all $n>(2s-1)^2$.

Upper bounds for the rank of powers of quadrics

TL;DR

The paper establishes tight asymptotic control on the Waring rank of powers of quadrics in many variables, showing while the generic rank scales as . It introduces a unifying framework based on partitions and monomial symmetric polynomials , enabling explicit decompositions and upper bounds, and proves that for all there is a threshold beyond which the rank is subgeneric, i.e., . The results hinge on catalecticant/apolarity techniques, symmetry-based decompositions, and asymptotic estimates for partition functions, providing both concrete decompositions for small and a general subgeneric-rank regime. These insights advance understanding of rank loci geometry and high-dimensional Waring problems in the context of quadratic powers.

Abstract

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any , we prove that the -th power of a quadratic form of rank grows as . Furthermore, we demonstrate that its rank is subgeneric for all .
Paper Structure (5 sections, 16 theorems, 144 equations)

This paper contains 5 sections, 16 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Powers of quadrics and classical decompositions
  3. Further examples of closed formulas
  4. Asymptotic growth and upper bounds for Waring rank
  5. Subgeneric rank

Key Result

Theorem 1

For every $n,s\in\mathbb{N}$ In particular, for any $s\in\mathbb{N}$, the rank of $q_n^s$ grows at most as $n^s$.

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4: BHMT18*Theorem 4.1
  • Corollary 5
  • Proposition 6: B. Reznick
  • Theorem 7
  • Definition 2.4
  • Definition 2.5
  • Example 8
  • ...and 18 more