Real Stability and Log Concavity are coNP-Hard

TL;DR

This paper proves that determining real stability and log concavity of fixed-degree polynomials is computationally hard (coNP-hard), but deciding Lorentzian property can be done efficiently in polynomial time.

Contribution

It establishes the computational complexity of deciding real stability, Lorentzian, and log concave properties of polynomials, revealing a complexity gap.

Findings

Deciding real stability is coNP-hard.

Deciding log concavity is coNP-hard.

Deciding Lorentzian property is polynomial-time solvable.

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.