Separating Cones Defined by Toric Varieties: Properties, Open Problems, and Application to Sums of Squares Representations of Polynomials

Core Concepts

This mathematics research paper explores a specific filtration of convex cones defined by projective varieties, aiming to refine Hilbert's 1888 theorem about the representation of positive semidefinite forms as sums of squares.

Abstract

• Bibliographic Information: Goel, C., Hess, S., & Kuhlmann, S. (2024). SEPARATING CONES DEFINED BY TORIC VARIETIES: SOME PROPERTIES AND OPEN PROBLEMS. arXiv preprint arXiv:2411.06468v1.

• Research Objective: This paper investigates a filtration of convex cones defined by projective varieties containing the Veronese variety. The authors aim to analyze the properties of these cones, particularly their closure, interior, and boundary, and to explore their potential in refining Hilbert's 1888 theorem on the representation of positive semidefinite forms as sums of squares.

• Methodology: The authors utilize the Gram matrix method to construct and analyze the cone filtration. They leverage algebraic geometry tools, including projective varieties, Veronese embeddings, and properties of quadratic forms, to study the geometric and algebraic characteristics of these cones.

• Key Findings: The paper demonstrates that all cones in the filtration are closed and provides explicit descriptions of their interiors and boundaries. It further establishes that none of the strictly separating cones in the filtration are spectrahedral shadows, implying limitations in using semidefinite programming for their analysis.

• Main Conclusions: The study deepens the understanding of the geometric and algebraic properties of the cones in the constructed filtration. It highlights the challenges in determining membership in these cones and their duals, posing open problems for further research. The authors suggest potential applications of their findings in polynomial optimization and the truncated moment problem.

• Significance: This research contributes to the field of real algebraic geometry, specifically to the study of positive semidefinite forms and their representations as sums of squares. It provides a refined understanding of the cone structures arising from this problem and opens avenues for developing more efficient algorithms for polynomial optimization.

• Limitations and Future Research: The paper acknowledges the lack of efficient membership tests for the studied cones and their duals. It proposes investigating intrinsic properties of representing Gram matrices and exploring connections to the truncated moment problem as potential directions for future research. Additionally, the authors suggest generalizing their approach to cones defined by toric varieties.

Stats

(n + 1, 2d) = (2, 2d)d⩾1 or (n + 1, 2)n⩾1 or (3, 4) represent the Hilbert cases.
k(n, d) + 1 represents the dimension of the vector space of forms in n + 1 variables of degree d.
The cone filtration is defined as Σn+1,2d = C0 ⊆. . . ⊆Cn ⊆Cn+1 ⊆. . . ⊆Ck(n,d)−n = 풫n+1,2d.

Quotes

Key Insights Distilled From

Separating Cones defined by Toric Varieties: Some Properties and Open Problems

by Charu Goel, ... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06468.pdf

Deeper Inquiries

How can the properties of the cone filtration be leveraged to develop more efficient algorithms for testing the positivity of polynomials, particularly in the context of optimization problems?

The cone filtration introduced in the paper provides a novel approach to testing the positivity of polynomials, a problem with significant implications for optimization. Here's how its properties can be leveraged for developing efficient algorithms:
1. Exploiting the Hierarchy: The cone filtration offers a hierarchy of increasingly stricter tests for polynomial positivity. Instead of directly checking if a polynomial is positive semidefinite (PSD), which is computationally demanding in general, one can first test its membership in the larger cones of the filtration. These initial tests can be designed to be computationally cheaper. If a polynomial fails to belong to a larger cone, it cannot be PSD, saving computational effort.
2. Focusing on Specific Cones: The paper identifies specific cones in the filtration, like those associated with varieties of minimal degree, where membership tests are more efficient. Algorithms can be tailored to exploit the structure of these cones. For instance, the paper highlights that membership in cones corresponding to varieties of minimal degree can be tested by checking intrinsic properties of representing Gram matrices, which can be done more efficiently than directly testing the positivity of the associated quadratic form.
3. Sparsity and Structure: The connection to the 풜-truncated moment problem opens avenues for exploiting sparsity in polynomial optimization problems. If the polynomial's support (the set of monomials with non-zero coefficients) is sparse, the corresponding 풜i-truncated moment problem might be significantly smaller than the full moment problem, leading to computational savings.
4. Alternative Characterizations: The paper acknowledges the limitations of semidefinite programming in characterizing the intermediate cones. This motivates the search for alternative characterizations, potentially through techniques like sums of squares modulo ideals or other algebraic-geometric methods, which could lead to more efficient membership tests.
Challenges and Future Directions:

Open Problem 1 in the paper directly addresses the need for efficient membership tests for the intermediate cones. Solving this problem is crucial for realizing the computational benefits of the cone filtration.
The connection to the 풜-truncated moment problem (Open Problem 4) is promising but requires further investigation to develop concrete algorithms.

Could there be alternative geometric constructions or different families of varieties that lead to a more computationally tractable characterization of positive semidefinite forms, potentially circumventing the limitations of the presented cone filtration?

The search for alternative geometric constructions and families of varieties is a promising direction for overcoming the limitations of the presented cone filtration. Here are some potential avenues:
1. Beyond Veronese Varieties: The current cone filtration relies on projective varieties containing the Veronese variety. Exploring other families of varieties with desirable properties, such as toric varieties (as hinted at in the paper), flag varieties, or determinantal varieties, could lead to different cone filtrations with potentially more tractable membership tests.
2. Relaxing the Containment Condition: The requirement that the varieties in the filtration contain the Veronese variety could be relaxed. Instead, one could consider families of varieties that intersect the Veronese variety in a structured way or have specific intersection properties with the sets where the polynomials are required to be positive.
3. Exploiting Symmetry: If the underlying polynomial optimization problem exhibits symmetries, incorporating these symmetries into the geometric construction could lead to a reduction in the complexity of the problem and more efficient characterizations.
4. Non-Commutative Settings: Generalizing the cone filtration to non-commutative settings, where polynomials are replaced by non-commutative operators, could be fruitful. This could connect to areas like free positivity and matrix inequalities, where geometric methods are actively being developed.
5. Numerical Algebraic Geometry: Techniques from numerical algebraic geometry, such as homotopy continuation and numerical optimization on varieties, could be employed to develop approximate but efficient membership tests for the cones.

What are the implications of this research for other areas of mathematics where the interplay between algebra and geometry is crucial, such as representation theory or algebraic topology?

The research presented in the paper, while focused on polynomial optimization, has potential implications for other areas of mathematics where the interplay between algebra and geometry is central:
1. Representation Theory:

Moment Maps and Orbit Closures: The cone filtration and the connection to the moment problem resonate with the study of moment maps in symplectic geometry and the geometry of orbit closures in representation theory. The varieties appearing in the filtration could potentially be related to such orbit closures, and the cone membership problem might translate into questions about the structure of these closures.
Non-Commutative Algebra: The generalization of the cone filtration to non-commutative settings, as mentioned earlier, could have direct implications for representation theory, particularly in the study of representations of non-commutative algebras and their associated geometric objects.
2. Algebraic Topology:

Betti Numbers and Homology: The geometric constructions used in the paper could potentially be analyzed using tools from algebraic topology. For instance, the Betti numbers of the varieties in the filtration might provide insights into the complexity of the cone membership problem.
Sheaf Cohomology: The study of sheaves and their cohomology on the varieties involved could offer a deeper understanding of the geometric properties of the cones and their relationship to polynomial positivity.
3. Real Algebraic Geometry:

Positive Polynomials and Sums of Squares: The paper directly contributes to the field of real algebraic geometry, particularly to the study of positive polynomials and sums of squares representations. The cone filtration provides a new perspective on the relationship between these concepts.
Semialgebraic Sets: The sets defined by polynomial inequalities, known as semialgebraic sets, are fundamental objects in real algebraic geometry. The cone filtration could potentially lead to new methods for studying the geometry and topology of these sets.