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Real Stability and Log Concavity are coNP-Hard

Tracy Chin

Abstract

Real-stable, Lorentzian, and log-concave polynomials are well-studied classes of polynomials, and have been powerful tools in resolving several conjectures. We show that the problems of deciding whether a polynomial of fixed degree is real stable or log concave are coNP-hard. On the other hand, while all homogeneous real-stable polynomials are Lorentzian and all Lorentzian polynomials are log concave on the positive orthant, the problem of deciding whether a polynomial of fixed degree is Lorentzian can be solved in polynomial time.

Real Stability and Log Concavity are coNP-Hard

Abstract

Real-stable, Lorentzian, and log-concave polynomials are well-studied classes of polynomials, and have been powerful tools in resolving several conjectures. We show that the problems of deciding whether a polynomial of fixed degree is real stable or log concave are coNP-hard. On the other hand, while all homogeneous real-stable polynomials are Lorentzian and all Lorentzian polynomials are log concave on the positive orthant, the problem of deciding whether a polynomial of fixed degree is Lorentzian can be solved in polynomial time.
Paper Structure (15 sections, 25 theorems, 44 equations, 1 figure)

This paper contains 15 sections, 25 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Log-Concave and Lorentzian Polynomials
  5. Hyperbolic Polynomials
  6. Stable Polynomials
  7. Hierarchy of Polynomials
  8. Computational Complexity of Stability
  9. Lorentzianity
  10. Computational Complexity of Log Concavity
  11. Log Concavity For Cubics
  12. Directional Log Concavity
  13. Log Concavity for Homogeneous Cubics
  14. Conclusion and Future Work
  15. Acknowledgements

Key Result

Theorem 2.2

If $f$ is a homogeneous polynomial in $n \geq 2$ variables of degree $d\geq 2$, then the following are equivalent for any $w\in{\mathbb R}^n$ satisfying $f(w) > 0$.

Figures (1)

  • Figure 1: Cones $L_\varepsilon \supseteq K$, both containing $e_0$. These cones are constructed so that $p > 0$ on $L_\varepsilon$ and $K$, so $p$ is hyperbolic with respect to $e_0$ if and only if it is hyperbolic with respect to $K$.

Theorems & Definitions (48)

  • Definition 2.1
  • Theorem 2.2: brändén2022lorentzian, Proposition 2.33
  • Definition 2.3: anari2018logconcave
  • Definition 2.4
  • Theorem 2.5: garding1959hyperbolic, Theorem 2
  • Example 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Theorem 3.1: saunderson2019certifying, Proposition 5.1
  • ...and 38 more