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Stubborn Polynomials

This paper characterizes stubborn polynomials—nonnegative polynomials with no odd power that is a sum of squares—on smooth and singular curves and ternary sextics, proving that their existence and properties depend on the reality of zeros, the curve's genus, and the real delta-invariant, thereby fully resolving a conjecture by Blekherman, Kozhasov, and Reznick.

Original authors: Lorenzo Baldi, Grigoriy Blekherman, Khazhgali Kozhasov, Daniel Plaumann, Bruce Reznick, Rainer Sinn

Published 2026-02-03

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CC BY 4.0

Original authors: Lorenzo Baldi, Grigoriy Blekherman, Khazhgali Kozhasov, Daniel Plaumann, Bruce Reznick, Rainer Sinn

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 are a mathematician trying to solve a puzzle involving shapes and numbers. In the world of Real Algebraic Geometry, there is a famous question: If a polynomial (a mathematical expression with variables like $x, y, z$ ) is always positive or zero (nonnegative) no matter what numbers you plug in, can you always prove it by writing it as a "Sum of Squares" (like $x^2 + y^2$ )?

Hilbert, a famous mathematician, showed long ago that the answer is no. Sometimes, a shape is positive, but you cannot build it out of simple squares.

This paper introduces a new character in this story: the "Stubborn Polynomial."

What is a "Stubborn" Polynomial?

Think of a stubborn polynomial as a very difficult guest at a party.

The Rule: The guest is always polite (nonnegative).
The Problem: You cannot describe them using the standard "Sum of Squares" recipe.
The Stubbornness: Even if you try to "boost" them by raising them to an odd power (like cubing them, or raising them to the 5th power), they still refuse to become a Sum of Squares. They remain uncooperative no matter how many times you try to force them into that shape.

The authors of this paper are trying to figure out: When does a polynomial become this stubborn?

The Main Discoveries

1. The "Real-Rooted" Rule for Smooth Curves

The authors looked at polynomials defined on smooth, curved lines (called curves). They found a simple rule for when a polynomial is stubborn:

The Metaphor: Imagine the polynomial is a net trying to catch fish. The "fish" are the points where the polynomial equals zero.
The Finding: If the net only catches real fish (points that actually exist in the real world, not imaginary ones), the polynomial is stubborn.
The Surprise: If the net catches even one "imaginary fish" (a complex number), the polynomial is not stubborn; it will eventually give up and become a Sum of Squares if you raise it to a high enough power.

This means that on smooth curves, being stubborn is exactly the same as having all your zeros be real numbers.

2. The "Zoo" of Broken Curves

What if the curve isn't smooth? What if it has a sharp point (a singularity) or is broken into pieces?

The authors created a "zoo" of examples using cubic curves (curves defined by degree 3 equations).
The Result: It's a mixed bag.
- Some broken curves (like a "nodal" cubic with a connected loop) still have stubborn polynomials.
- Other broken curves (like a "cuspidal" cubic that looks like a sharp spike) have no stubborn polynomials at all. No matter how hard you try, every nonnegative polynomial on these shapes will eventually become a Sum of Squares.
- Some curves only have stubborn polynomials in very high degrees (like degree 4 or higher), but not in low degrees.

3. The Case of the "Ternary Sextic" (The 6th Degree Puzzle)

The paper focuses heavily on a specific type of polynomial: Ternary Sextics.

Translation: These are polynomials with 3 variables ( $x, y, z$ ) and a total degree of 6.
The Conjecture: The authors proved a guess made by their colleagues. They found a "magic number" that determines stubbornness.
The Magic Number: They count the "real zeros" of the polynomial, but they weigh them based on how sharp or complex the zero is (using something called the real delta-invariant).
- If this count is 9 or higher, the polynomial is Stubborn.
- If the count is 8 or lower, the polynomial is Not Stubborn (it will eventually become a Sum of Squares).

Think of it like a weight limit. If the polynomial has "enough weight" in its real zeros (at least 9), it's too heavy to be forced into the Sum of Squares shape. If it's too light (8 or less), it can be forced.

4. Lifting: Building Tall Towers

The authors also studied how to take a stubborn polynomial from a small curve and "lift" it up to a bigger, multi-dimensional space (like taking a 2D drawing and making it a 3D sculpture).

The Finding: If you have a stubborn polynomial on a specific type of curve (an "elliptic normal curve"), you can often lift it to a larger space and it will remain stubborn.
The Analogy: Imagine you have a stubborn knot on a string. If you tie that string to a larger, more complex structure in a specific way, the knot stays stubborn. This allows them to create stubborn polynomials in higher dimensions by starting with simple ones on curves.

Summary

This paper is a map of the "Stubborn Landscape."

On smooth curves: A polynomial is stubborn if and only if all its zeros are real numbers.
On broken curves: It depends on the shape. Some shapes allow stubbornness; others ban it completely.
On 3D shapes (Ternary Sextics): There is a precise threshold (a count of 9) that separates the stubborn from the non-stubborn.
The Method: They used advanced geometry (like blowing up surfaces to smooth them out) and the theory of "weak Del Pezzo surfaces" (a specific type of geometric shape) to prove these rules.

In short, the paper tells us exactly when a positive polynomial refuses to be a Sum of Squares, turning a vague mathematical mystery into a set of clear, countable rules.

Technical Summary of "Stubborn Polynomials"

Problem Statement
The paper investigates the relationship between nonnegative polynomials and sums of squares (SOS) on real projective varieties, a central topic in real algebraic geometry. While Hilbert established the existence of nonnegative polynomials that are not SOS, and Artin resolved Hilbert's 17th problem by showing they are SOS of rational functions, this work focuses on an intermediate certificate of nonnegativity. Specifically, the authors study "stubborn" polynomials: nonnegative polynomials $f$ on a real variety $X$ such that no odd power $f^{2k+1}$ is a sum of squares of polynomials. Previously, such polynomials were studied primarily in the globally nonnegative case (on the ambient space $\mathbb{P}^n$ ) with limited results restricted to $\mathbb{P}^2$ . The authors aim to fully characterize stubborn polynomials on real projective curves and surfaces, exploring how their existence and properties relate to the geometry of their real zero sets and the underlying variety.

Methodology
The authors employ a combination of real algebraic geometry, the theory of Jacobians and generalized Jacobians, and the geometry of surfaces.

Curve Theory: For smooth curves, the analysis relies on the generalized Jacobian and the properties of 2-torsion points. The authors utilize results from Scheiderer regarding strictly positive forms and the structure of the cone of nonnegative forms.
Singular and Reducible Curves: The study of singular curves involves explicit computations for plane cubics and the analysis of 2-torsion points in the generalized Jacobian of singular curves.
Surface Theory (Ternary Sextics): The classification of stubborn ternary sextics (degree 6 forms on $\mathbb{P}^2$ ) utilizes the theory of weak Del Pezzo surfaces. The authors perform embedded resolutions of singularities by iterated blow-ups to transform the problem into one involving strictly positive sections on a surface. They apply Scheiderer's Positivstellensatz on these surfaces to determine when odd powers become SOS.
Lifting Techniques: A key methodological tool is the "lifting" of nonnegative forms from a curve to a higher-dimensional variety. The authors analyze conditions under which a nonnegative quadratic form on a curve can be extended to a nonnegative quadratic form on the ambient variety, utilizing tangent space conditions and Veronese embeddings.

Key Contributions and Results

Characterization on Smooth Curves:
The paper provides a complete characterization of stubborn polynomials on smooth, totally real projective curves. A nonnegative form $F$ on such a curve is stubborn if and only if it is real-rooted (all its zeros on the curve are real) and not a sum of squares.
- Implication: This implies that for smooth curves of genus $g \ge 1$ , stubborn polynomials exist in all sufficiently high degrees. Conversely, on smooth rational curves (genus 0), no stubborn polynomials exist in any degree.
- Surprising Consequence: Unlike the conjectured behavior for $\mathbb{P}^2$ , where every extreme ray of the nonnegative cone (that is not a square) is stubborn, on smooth curves only specific extreme rays (those with maximal real zeros) are stubborn.
Existence on Singular and Reducible Curves:
The authors explore the "zoo" of plane cubics (smooth, nodal, cuspidal, reducible). They demonstrate that the existence of stubborn polynomials on singular curves is delicate.
- Some singular curves (e.g., cuspidal cubics or nodal cubics with solitary nodes) admit no stubborn polynomials of any degree.
- Others admit stubborn forms only in specific degrees (e.g., stubborn quartics exist on certain reducible configurations where no stubborn quadrics exist).
- The existence is linked to the presence of non-trivial 2-torsion points in the generalized Jacobian of the curve.
Classification of Ternary Sextics:
The paper proves a conjecture by Blekherman, Kozhasov, and Reznick regarding ternary sextics (degree 6 forms on $\mathbb{P}^2$ ).
- Theorem: A nonnegative ternary sextic $F$ (which is not a sum of squares) is stubborn if and only if its real delta-invariant $\delta_R(F)$ is at least 9.
- Method: If $\delta_R(F) \le 8$ , the authors construct a weak Del Pezzo surface via blow-ups where the strict transform of the sextic becomes strictly positive, allowing the application of Scheiderer's theorem to show an odd power is SOS. If $\delta_R(F) \ge 9$ , the form inherits stubbornness from its restriction to a cubic curve.
Lifting Nonnegativity:
The authors develop criteria for "lifting" nonnegative forms from a curve $C$ to a higher-dimensional variety $X$ .
- They prove that if $X$ is a totally real variety defined by quadrics and $C$ is an elliptic normal curve satisfying specific tangent space conditions ( $T_p C = T_p X \cap L$ ), then any nonnegative quadratic form on $C$ can be extended to a nonnegative quadratic form on $X$ .
- This lifting preserves stubbornness, allowing the construction of stubborn polynomials on higher-dimensional varieties with controlled numbers of real zeros.

Significance and Claims
The paper claims to significantly improve the understanding of the relationship between the geometry of real zeros and the algebraic property of being a sum of squares (or an odd power thereof).

Structural Insight: It establishes that for smooth curves, stubbornness is equivalent to real-rootedness, a stark contrast to the global case on $\mathbb{P}^2$ .
Completeness: It provides a full classification of stubborn ternary sextics, resolving a specific conjecture and demonstrating that the threshold for stubbornness is exactly 9 real delta-invariants.
New Constructions: By combining curve results with lifting techniques, the authors demonstrate the existence of stubborn polynomials on higher-dimensional varieties (specifically on $\mathbb{P}^n$ ) with "few" real zeros, challenging previous intuitions about the scarcity of such forms.
Open Directions: The work opens new research directions regarding the classification of varieties with no stubborn quadrics and the behavior of stubborn polynomials on singular varieties, noting that the existence of stubborn forms is not guaranteed even for smooth curves in low degrees.

The authors maintain a modest tone regarding future implications, framing their results as foundational steps toward understanding the "zoo" of real algebraic varieties where nonnegativity does not imply sums of squares, and where the "stubborn" property acts as a specific geometric marker.