Determining the Computational Complexity of Stability and Log Concavity for Polynomials

Core Concepts

Determining the computational complexity of deciding whether a polynomial is real stable, completely log concave, or log concave on the nonnegative orthant.

Abstract

The content analyzes the computational complexity of testing various properties of polynomials, including real stability, complete log concavity, and log concavity.

Key highlights:

1. Deciding whether a homogeneous cubic polynomial is real stable is coNP-hard.
2. Deciding whether a homogeneous polynomial of fixed degree is completely log concave can be done in polynomial time.
3. Deciding whether a homogeneous polynomial of degree 4 or higher is log concave on the nonnegative orthant is coNP-hard.
4. For homogeneous cubic polynomials, deciding log concavity can be done in polynomial time, but deciding directional log concavity is coNP-hard.

The content provides a detailed analysis of the hierarchy of stable, Lorentzian, and log-concave polynomials, and establishes the computational complexity of testing these properties. The results shed light on the inherent difficulty of working with these important classes of polynomials.

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Key Insights Distilled From

Real Stability and Log Concavity are coNP-Complete

by Tracy Chin at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00162.pdf

Deeper Inquiries

Are there any other properties of polynomials that have interesting computational complexity results

One property of polynomials that has interesting computational complexity results is the property of being hyperbolic. Hyperbolic polynomials have connections to optimization problems and have been studied extensively in the field of mathematics. The computational complexity of testing whether a polynomial is hyperbolic with respect to a given direction has been shown to be coNP-hard, indicating that this property is not easy to determine algorithmically. This complexity result adds to the understanding of the computational challenges involved in analyzing the hyperbolicity of polynomials.

What are the implications of these complexity results for practical applications that rely on stable, Lorentzian, or log-concave polynomials

The computational complexity results for stable, Lorentzian, and log-concave polynomials have significant implications for practical applications that rely on these properties. For example, in optimization problems, the stability of polynomials is crucial for ensuring the convergence and efficiency of algorithms. The coNP-completeness of testing real stability and log concavity implies that these properties are not easy to verify computationally, which can impact the development and implementation of optimization algorithms.
In applications related to combinatorial optimization, the properties of Lorentzian and log-concave polynomials play a key role in modeling and solving various problems. The coNP-hardness of testing log concavity for homogeneous cubics highlights the computational challenges in analyzing these types of polynomials. Understanding the complexity of verifying these properties can guide the design of algorithms and computational methods in combinatorial optimization and related fields.
Overall, the computational complexity results for stable, Lorentzian, and log-concave polynomials provide insights into the difficulty of analyzing these properties algorithmically, which can influence the development and application of mathematical models and optimization techniques in various domains.

Are there any connections between the computational complexity of testing these polynomial properties and the underlying mathematical structures they represent

The computational complexity of testing properties such as stability, Lorentzianity, and log concavity of polynomials is closely related to the underlying mathematical structures they represent. These properties are not only important in mathematical theory but also have practical implications in various fields such as optimization, combinatorial optimization, and computational geometry.
The connections between the computational complexity of testing these polynomial properties and their mathematical structures lie in the intricate relationships between the coefficients, degrees, and geometric properties of the polynomials. For example, the coNP-completeness of testing real stability and log concavity indicates the complexity of analyzing the behavior of polynomials in optimization problems and geometric settings.
Furthermore, the coNP-hardness of testing log concavity for homogeneous cubics reveals the challenges in characterizing the curvature and convexity properties of these polynomials. Understanding the computational complexity of these tests provides insights into the underlying mathematical structures of polynomials and their applications in diverse areas of mathematics and computer science.