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On Waring rank jumps via critical rank-one approximations

This paper investigates how eigenvectors (critical rank-one approximations) of symmetric tensors influence their Waring rank, demonstrating that for binary forms, such eigenvectors generally increase the rank unless they are already part of a minimal decomposition, with specific codimension results established for degree-d rank-r forms.

Original authors: Alessandro Oneto, Pierpaola Santarsiero, Ettore Teixeira Turatti

Published 2026-05-07

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CC BY 4.0

Original authors: Alessandro Oneto, Pierpaola Santarsiero, Ettore Teixeira Turatti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a complex, multi-layered sculpture made of clay. In the world of mathematics, this sculpture is called a symmetric tensor (or a "form"). One way to understand this sculpture is to try to build it by stacking up simple, single-layer shapes (like flat sheets or simple curves) on top of each other.

The Waring rank is simply the minimum number of these simple shapes you need to stack to recreate your complex sculpture exactly. If you can build it with 3 shapes, its rank is 3. If you need 10, its rank is 10. Finding this minimum number is notoriously difficult, like trying to find the shortest path through a maze with billions of dead ends.

Now, imagine you have a special tool called an eigenvector. In this context, think of an eigenvector as a "critical point" or a "sweet spot" on your sculpture. It's a specific direction where the shape behaves in a very predictable, balanced way. Mathematicians have long wondered: If I take one of these "sweet spot" shapes and try to use it to build my sculpture, will it help me get closer to the minimum number of pieces (lower the rank), or will it mess things up and force me to use more pieces (raise the rank)?

This paper investigates exactly that question. Here is the breakdown of their findings using everyday analogies:

1. The Two Main Questions

The authors are looking at two opposite scenarios:

The "Helper" Scenario: Can we find a "sweet spot" shape that is already part of the most efficient construction plan? If we use it, can we subtract it and be left with a simpler sculpture that needs fewer pieces?
The "Hindrance" Scenario: If we grab a "sweet spot" shape that isn't part of the best plan, does adding it or trying to use it make the sculpture harder to build, forcing us to use more pieces than necessary?

2. The "Binary" Case (The Simplest Sculptures)

The paper focuses heavily on "binary forms," which are like sculptures made with only two types of ingredients (think of them as 2D shapes rather than 3D ones).

When the sculpture is "subgeneric" (not too complex):
If the sculpture is relatively simple (specifically, if its rank is less than half the degree of the shape plus one), the authors found a surprising rule: Almost every "sweet spot" shape you pick will actually make the problem harder.
- The Analogy: Imagine you are trying to build a house with the fewest bricks possible. If you pick a "perfectly balanced" brick that isn't part of the optimal blueprint, trying to force it into the wall will actually make the house unstable, requiring you to add more bricks to fix it. The paper proves that for simple sculptures, these "sweet spots" are almost always traps that increase the rank.
When the sculpture is "identifiable" (has a unique blueprint):
For certain specific types of sculptures where there is only one way to build them, the authors mapped out exactly when a "sweet spot" is actually part of the blueprint. They found that this happens very rarely. It's like finding a specific key that fits a specific lock; most keys (eigenvectors) won't open the door to a simpler solution.

3. The "Monomial" Case (The Perfectly Symmetric Shapes)

The authors also looked at "monomials," which are like perfectly symmetrical shapes (e.g., $x^3$ or $x^2y$ ).

They discovered that for these shapes, the "sweet spots" and the "building blocks" overlap in a very specific, predictable way. In fact, for these shapes, you can almost always find a "sweet spot" that is part of the minimal construction plan. It's like a perfectly round ball where every direction is a "sweet spot," and you can easily find one that fits the blueprint.

4. The "Critical Waring Variety" (The Map of Exceptions)

The authors created a mathematical "map" (called the Critical Waring Variety) to show where these "helpful" eigenvectors exist.

They found that for binary shapes, this map is a very thin slice (a "codimension-one" slice) of the total space of possibilities.
The Analogy: Imagine a vast ocean of all possible sculptures. The places where a "sweet spot" actually helps you build the sculpture more efficiently are like a very thin, invisible sheet of glass floating in that ocean. If you pick a sculpture at random, you are unlikely to land on that sheet. You are much more likely to land in the "water" where the eigenvectors will only increase the rank.

5. The "Rank Jump" (When Things Get Worse)

The paper concludes with a strong warning for the "subgeneric" cases (the simpler sculptures).

If you have a simple sculpture and you try to subtract a "sweet spot" shape from it, the rank will almost certainly go up.
The Analogy: It's like trying to fix a leaky boat by removing a specific plank that looks perfectly balanced. Instead of fixing the leak, you create a bigger hole, and now you need more wood to patch it up. The paper proves that for generic simple shapes, this "rank jump" is the norm, not the exception.

Summary

In simple terms, this paper is a study of efficiency vs. traps.

The Trap: For most simple, complex shapes, the "special" directions (eigenvectors) that look promising are actually traps. Using them doesn't simplify the shape; it complicates it, forcing you to use more building blocks.
The Rare Exception: There are very specific, rare cases (like perfectly symmetrical shapes or very specific complex arrangements) where a "sweet spot" is part of the most efficient blueprint.
The Takeaway: You cannot assume that just because a shape has a "balanced" direction (an eigenvector), it will help you simplify the shape. In fact, for most simple shapes, it will do the exact opposite.

Technical Summary: On Waring Rank Jumps via Critical Rank-One Approximations

Problem Statement
The paper investigates the relationship between the eigenvectors (also termed critical rank-one approximations) of a symmetric tensor and its Waring rank (the minimal length of a decomposition into powers of linear forms). While singular pairs in matrices provide a direct path to minimal additive decompositions, this correspondence fails for higher-order tensors. The central problem addressed is whether an eigenvector of a symmetric tensor $F$ can be utilized to either decrease its Waring rank (by being part of a minimal decomposition) or increase it (by subtracting the eigenvector term). Specifically, the authors analyze the intersection of the Waring locus $W(F)$ (linear forms appearing in minimal decompositions) and the eigenscheme $Eig(F)$ (the set of all eigenvectors).

Methodology
The authors employ a geometric and algebraic approach grounded in the interplay between the apolar action and the Bombieri-Weyl product.

Apolarity and Bombieri-Weyl Product: A key technical observation is that the Bombieri-Weyl product (used to define eigenvectors via orthogonality to the tangent space of the Veronese variety) is proportional to the apolar action (differentiation). This allows the translation of eigenvector conditions into conditions on apolar ideals.
Geometric Framework: The study utilizes secant varieties of the Veronese variety to represent forms of a given rank. The authors define the "critical Waring variety" $WEd_{n,r}$ as the Zariski closure of forms where $W(F) \cap Eig(F) \neq \emptyset$ .
Binary Forms and Monomials: The analysis focuses heavily on binary forms ( $n=1$ ) and monomials, where the structure of apolar ideals is well-understood (generated by two forms). The authors use the characterization of forbidden loci and Waring loci for these specific cases to derive general properties.

Key Contributions and Results

Characterization of Eigenvectors in Minimal Decompositions:
- The paper characterizes the intersection $W(F) \cap Eig(F)$ using apolar ideals. For a form $F$ with rank equal to the number of essential variables, a summand in a minimal decomposition is an eigenvector if and only if it is orthogonal to all other summands (Proposition 3.4).
- For binary forms of rank 3, the authors prove that if a linear form is both in the Waring locus and the eigenscheme, its coefficient in the minimal decomposition must differ from its singular value (Proposition 3.7). Consequently, subtracting the eigenvector with its singular value does not reduce the rank.
The Critical Waring Variety ( $WEd_{n,r}$ ):
- The authors study the variety of forms admitting an eigenvector in their minimal decomposition.
- Binary Forms: For binary forms of rank $r \leq (d+1)/2$ , $WEd_{1,r}$ is shown to be an irreducible subvariety of codimension one within the $r$ -th secant variety of the rational normal curve (Theorem 4.10, Theorem 4.12).
- General Forms: For $n \geq 2$ , the paper establishes that $WEd_{n,r}$ has codimension at most $n$ in the $r$ -th secant variety of the Veronese variety (Theorem 4.14).
- The paper provides explicit equations for the locus where a specific linear form is an eigenvector of a rank- $r$ form (Theorem 4.3).
Rank Increase via Eigenvectors:
- The paper addresses the converse: when does an eigenvector increase the rank?
- Subgeneric and Generic Ranks: For binary forms of subgeneric rank ( $r < (d+1)/2$ ) and for generic forms of generic rank in odd degrees, the authors prove that any eigenvector increases the Waring rank (Corollary 5.3). This is because, in these cases, the eigenscheme is generically disjoint from the Waring locus and contained entirely within the forbidden locus.
- Monomials: For monomials with $n \geq 2$ , the intersection of the eigenscheme and the Waring locus is always non-empty, implying that eigenvectors can sometimes be used to decrease the rank (Theorem 3.16).
Specific Examples and Counterexamples:
- The authors provide examples of binary forms where the intersection $W(F) \cap Eig(F)$ is non-empty, demonstrating that eigenvectors can be part of minimal decompositions (Example 3.9).
- Conversely, they show cases where eigenvectors increase the rank, confirming and extending results from [SC10] regarding real binary cubics to the complex setting and higher degrees.

Significance and Claims
The paper claims to clarify the often ambiguous relationship between eigenvectors and minimal Waring decompositions. It demonstrates that while eigenvectors are critical points for rank-one approximation, they are typically not part of a minimal decomposition for generic forms.

The work establishes that for "most" binary forms (subgeneric and generic odd-degree), eigenvectors are strictly "forbidden" for rank reduction and instead serve to increase the rank.
The authors identify specific geometric loci (the critical Waring variety) where the "ideal" behavior of matrices (where eigenvectors align with minimal decompositions) occurs for tensors, showing this is a codimension-one phenomenon for binary forms.
The paper concludes that the relation between the eigenscheme and the forbidden locus for suprageneric rank forms remains an open area requiring further investigation, particularly for even degrees where the generic rank behavior differs from the odd-degree case.

The authors explicitly state that their approach relies on the strict relation between apolar action and the Bombieri-Weyl product to translate geometric questions about the Veronese variety into algebraic questions about apolar ideals. They do not propose new algorithms for computing Waring rank but rather provide theoretical bounds and structural characterizations of when eigenvectors can or cannot be used to modify the rank.