[Paper Review] Preserving positive polynomials and beyond

Julius Borcea, Alexander Guterman+1 more|ArXiv.org|Jan 11, 2008

Polynomial and algebraic computation11 references3 citations

TL;DR

This paper investigates linear operators—particularly finite and infinite-order ordinary differential operators with constant coefficients—that preserve positivity, non-negativity, or ellipticity in univariate polynomials. It proves that no finite-order differential operator preserves these positivity classes beyond degree 2k, while infinite-order operators with suitably positive coefficient sequences do preserve them, characterized via Hankel matrix definiteness and moment conditions on coefficients.

ABSTRACT

Following the classical approach of Pólya-Schur theory we initiate in this paper the study of linear operators acting on $\mathbb{R}[x]$ and preserving either the set of positive univariate polynomials or similar sets of non-negative and elliptic polynomials.

Motivation & Objective

• To classify linear operators on ℝ[x] that preserve positivity, non-negativity, or ellipticity in univariate polynomials.
• To extend Pólya-Schur theory beyond hyperbolicity-preserving operators to non-negative and positive polynomials.
• To determine the structural limitations of finite-order differential operators in preserving positivity classes.
• To characterize infinite-order differential operators with constant coefficients that preserve positivity and non-negativity.
• To establish necessary and sufficient conditions on coefficient sequences using Hankel matrix definiteness.

Proposed method

• Analyzes linear ordinary differential operators of finite and infinite order with polynomial or constant coefficients.
• Applies the Pólya-Schur framework to study linear preservers of positivity, non-negativity, and ellipticity in ℝ[x].
• Uses the action of operators on constant and linear terms to derive necessary conditions for preservation.
• Employs Hankel matrices formed from scaled coefficients (i!α_i) to characterize positivity and non-negativity preservation.
• Leverages shift-invariance of constant-coefficient operators to relate values at different points via translation.
• Reduces the problem to moment conditions on coefficients: ∑ i!a_i α_i ≥ 0 for non-negative polynomials.

Experimental results

Research questions

• RQ1 Which finite-order linear differential operators with polynomial coefficients preserve the set of non-negative polynomials in ℝ[x]?
• RQ2 Can any finite-order differential operator preserve positivity beyond degree 2k for a k-th order operator?
• RQ3 What conditions on the coefficient sequence α = (α_0, α_1, ...) ensure that an infinite-order constant-coefficient differential operator preserves non-negativity in ℝ_k[x]?
• RQ4 How are the preservation of positivity and the definiteness of Hankel matrices of scaled coefficients (i!α_i) related?
• RQ5 Is there a structural gap between the behavior of finite-order and infinite-order differential operators in preserving positivity classes?

Key findings

• No finite-order linear differential operator with polynomial coefficients preserves the set of non-negative, positive, or elliptic polynomials in ℝ[x], even in degree 2k.
• For any finite-order operator of order k ≥ 1, there exists a non-negative polynomial of degree 2k whose image under the operator is not non-negative.
• Infinite-order constant-coefficient differential operators preserve positivity (resp. non-negativity) in ℝ_k[x] if and only if the Hankel matrix formed from (i!α_i) is positive definite (resp. positive semi-definite).
• The coefficient condition ∑_{i=0}^k i!a_i α_i ≥ 0 for all non-negative polynomials ∑ a_i x^i is necessary and sufficient for non-negativity preservation in ℝ_k[x].
• The action of an infinite-order operator on ℝ_k[x] coincides with its truncation to order k, enabling the characterization of infinite-order operators via finite-dimensional Hankel matrices.
• There is no finite-order constant-coefficient operator that preserves positivity in ℝ_l[x] for any l > k, even if it works in ℝ_k[x], due to the existence of high-degree perturbations that violate positivity.