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Copositivity, discriminants and nonseparable signed supports

This paper establishes a connection between discriminants and copositivity of signomials by providing a criterion based on the intersection of a sign-preserving path with a signed discriminant, demonstrating that for nonseparable signed supports, copositivity can be decided via a single homotopy path and that such copositive polynomials decompose into sums of nonnegative circuit polynomials.

Original authors: Elisenda Feliu, Joan Ferrer, Máté L. Telek

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Elisenda Feliu, Joan Ferrer, Máté L. Telek

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 chef trying to bake a cake that is guaranteed to be sweet (non-negative) no matter how you slice it or where you take a bite. In the mathematical world, these "cakes" are called polynomials or signomials, and the "sweetness" is called copositivity.

The paper you provided is like a new, high-tech recipe book that helps chefs determine if their cake will always be sweet, without having to taste every single possible bite. Here is how the authors break it down, using simple analogies.

1. The Problem: The Infinite Tasting Test

Usually, to prove a cake is sweet everywhere, you'd have to taste it at every single point in the kitchen. That's impossible. Mathematicians have tried other tricks, like checking if the cake is made of "sweet ingredients" (sums of squares), but sometimes those tricks fail.

The authors focus on a specific type of cake: sparse signomials. Think of these as cakes with very specific, limited ingredients (monomials) where the "flavor" (exponents) can be any real number, not just whole numbers. The goal is to figure out: Is this specific recipe guaranteed to be sweet everywhere?

2. The New Tool: The "Discriminant" as a Danger Zone

The authors introduce a concept called the signed discriminant. Imagine the kitchen is a giant map.

The Safe Zone: This is where your cake recipe is guaranteed to be sweet.
The Danger Zone (The Discriminant): This is a thin, invisible wall on the map. If your recipe touches this wall, the cake has a "singular" point—a place where it stops being sweet and starts getting bitter, or where the texture changes drastically.

The paper's main discovery is a way to navigate this map. Instead of tasting the cake, you draw a path from your current recipe toward a "standard" recipe.

If your path hits the Danger Zone before you reach a certain checkpoint (specifically, before you reach the "1" mark on your path), your cake is not guaranteed to be sweet.
If your path hits the Danger Zone after that checkpoint (or never hits it), your cake is guaranteed to be sweet.

This is a huge shortcut. You don't need to taste the cake; you just need to calculate where your path crosses that invisible wall.

3. The Special Case: "Non-Separable" Supports

Sometimes, the Danger Zone is a messy, tangled knot, making it hard to find exactly where your path crosses it. The authors identify a special class of recipes called non-separable signed supports.

The Analogy: Imagine your ingredients are scattered on a table.

Separable: The "bad" ingredients (negative signs) are stuck in a corner, separated from the "good" ones. This makes the math messy and hard to solve.
Non-Separable: The "bad" ingredients are nestled right in the middle of the "good" ones, surrounded by them.

The paper proves that if your ingredients are non-separable (the bad ones are surrounded by the good ones), the Danger Zone becomes very well-behaved.

The Magic: There is only one single point where your path crosses the Danger Zone.
The Benefit: Instead of searching for a needle in a haystack, you only need to follow one single thread (a mathematical path) to find that exact crossing point. This makes the calculation incredibly fast and reliable.

4. The "SONC" Connection: Building with Lego Blocks

The paper also connects this to a method called SONC (Sum of Nonnegative Circuits).

Think of a complex cake as a structure built from small, simple, guaranteed-sweet Lego blocks (circuits).
Usually, you can't always build a complex cake out of these simple blocks.
The Breakthrough: The authors prove that for all those "non-separable" recipes (where the bad ingredients are surrounded), you can always build the cake out of these simple, sweet Lego blocks. If it's sweet, it's made of sweet blocks. If it's made of sweet blocks, it's sweet.

5. The Computer Implementation

Finally, the authors built a computer program (a Julia package) to do this work.

The Old Way: To check if a cake is sweet, you might have to check millions of paths or taste millions of points.
The New Way: For the "non-separable" cakes, the program just follows one single path to find the crossing point.
The Result: It is much faster and more accurate, especially for cakes that are almost bitter (very close to the Danger Zone). Other methods might get confused and say a bitter cake is sweet, but this new method can tell the difference even when the bitterness is tiny.

Summary

In short, this paper gives mathematicians a new, efficient way to check if a specific type of mathematical function is always positive.

It uses a "path" to find a "danger wall" (discriminant).
If the ingredients are "non-separable" (bad ones surrounded by good ones), there is only one place to look for that wall.
This allows for a super-fast, single-path calculation that is more accurate than previous methods, proving that these functions can always be built from simple, guaranteed-positive pieces.

Technical Summary: Copositivity, Discriminants and Nonseparable Signed Supports

Problem Statement
The paper addresses the problem of determining the copositivity of signomials—functions defined on the positive real orthant $\mathbb{R}^n_{>0}$ as linear combinations of monomials with real exponents. Specifically, it investigates the relationship between the boundary of the cone of copositive signomials, discriminants, and the combinatorial structure of the signomial's signed support (the set of exponents paired with the signs of their coefficients). While global nonnegativity is a central topic in real algebraic geometry, deciding membership in the cone of copositive polynomials is NP-hard. Existing methods often rely on certificates of nonnegativity, such as Sums of Squares (SOS), Sums of Nonnegative Circuits (SONC), or Sums of Arithmetic–Geometric Exponentials (SAGE). A key open question is characterizing the signed supports for which every copositive signomial admits a SONC decomposition.

Methodology
The authors develop a criterion for copositivity that relies on the geometry of the signed discriminant and homotopy continuation methods.

Signed Discriminants and Paths: The authors define the signed $(A^+, A^-)$ -discriminant, $\nabla_{>0}(A^+, A^-)$ , as the set of coefficient vectors for which the signomial (or its truncations to faces of the convex hull of the support) possesses a positive singular zero. They construct a specific path in the coefficient space, $c(t)$ , which scales the coefficients of the negative terms by a parameter $t$ while preserving the signs of the coefficients.
Intersection Criterion: The core theoretical tool is the observation that a signomial is copositive if and only if the path $c(t)$ does not intersect the signed discriminant for any $t < 1$ . The set of intersection points $T$ is analyzed to find a minimum value $t^*$ . Copositivity is equivalent to $t^* \geq 1$ .
Nonseparable Signed Supports: The paper introduces the combinatorial concept of "nonseparable" signed supports. A pair $(A^+, A^-)$ is nonseparable if $A^-$ is contained within a cell $D$ of the common refinement of all regular polyhedral subdivisions of $A^+$ . This geometric condition generalizes previously known cases where SONC certificates are guaranteed.
Homotopy Continuation: For nonseparable supports, the authors prove that the set $T$ contains exactly one element $t^*$ . This allows the use of parameter homotopy continuation to track a single solution path from a known starting point (where the signomial has a singular zero at $x=1$ ) to the target signomial, avoiding the need to solve multiple systems or evaluate critical points directly.

Key Contributions and Results

Copositivity Criterion (Theorem A/Theorem 2.9): The authors establish a necessary and sufficient condition for copositivity. For a signomial with signed support $(A^+, A^-)$ where $A^- \subseteq \text{conv}(A^+)$ , copositivity holds if and only if the minimum parameter $t^*$ at which the path $c(t)$ intersects the signed discriminant (or the discriminants of its truncations) satisfies $t^* \geq 1$ . This criterion avoids the numerical errors associated with evaluating the signomial at critical points.
SONC Decomposition for Nonseparable Supports (Theorem B/Theorem 3.16): The paper proves that for any nonseparable signed support where $A^-$ lies in the relative interior of a face of $\text{conv}(A^+)$ , the cone of copositive signomials coincides with the SONC cone. In other words, every copositive signomial with such a support admits a decomposition into a sum of nonnegative circuit signomials. This generalizes previous results that required specific geometric conditions (e.g., simple vertices).
Uniqueness and Nondegeneracy (Theorem C/Theorem 3.17 & 4.5): For nonseparable supports, the intersection set $T$ consists of exactly one point $t^*$ . Furthermore, the corresponding critical system has a unique, nondegenerate positive singular zero. This theoretical result ensures that numerical methods (homotopy continuation) are robust and efficient for this class of problems, as they do not suffer from missing solutions due to positive-dimensional zero sets.
Boundary Characterization (Theorem 3.18): The authors show that for nonseparable signed supports, the algebraic boundary of the cone of copositive polynomials is contained within (and equals, under specific conditions) the signed discriminant.
Computational Implementation: The paper presents the Julia package CopositivityDiscriminants.jl, which implements these methods. It demonstrates that for nonseparable supports, tracking a single homotopy path is significantly more efficient (seconds vs. minutes) and numerically stable compared to tracking all paths or using SAGE certificates, particularly for signomials very close to the discriminant boundary.

Significance and Claims
The paper claims to provide a unified framework connecting discriminants, copositivity, and SONC decompositions. Its primary significance lies in:

Generalization: It extends the known class of signed supports for which copositivity implies SONC decomposability to the broader class of nonseparable supports, removing restrictive assumptions like the existence of simple vertices.
Algorithmic Efficiency: By proving that nonseparable supports yield a unique, nondegenerate intersection with the discriminant, the authors enable a highly efficient computational strategy (single-path homotopy) that avoids the combinatorial explosion of solving multiple critical systems.
Numerical Stability: The method offers a certificate of copositivity that is more robust than evaluation-based methods or SAGE certificates when dealing with signomials near the boundary of the copositive cone, as demonstrated by examples where SAGE fails for small perturbations while the discriminant-based method succeeds.

The authors note that while the method is powerful for nonseparable supports, it faces limitations in the general case where the zero sets of critical systems may have positive dimensions, potentially leading to missed solutions in numerical implementations. However, they show that for generic coefficients, these pathologies do not occur.