[Paper Review] An extension of the Hermite-Biehler theorem with application to polynomials with one positive root

Richard Ellard, Helena Šmigoc|arXiv (Cornell University)|Jan 27, 2017

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper extends the Hermite-Biehler Theorem to non-Hurwitz stable polynomials by establishing a lower bound on the number of real roots of $ p(-x^2) $ and $ q(-x^2) $, showing these roots interlace based on the imbalance between roots in the left and right half-planes. It further applies this extension to derive an inverse rule of signs for polynomials with exactly one positive root, proving that their even and odd coefficient sequences each feature at most one sign change.

ABSTRACT

If a real polynomial $f(x)=p(x^2)+xq(x^2)$ is Hurwitz stable (every root if $f$ lies in the open left half-plane), then the Hermite-Biehler Theorem says that the polynomials $p(-x^2)$ and $q(-x^2)$ have interlacing real roots. We extend this result to general polynomials by giving a lower bound on the number of real roots of $p(-x^2)$ and $q(-x^2)$ and showing that these real roots interlace. This bound depends on the number of roots of $f$ which lie in the left half plane. Another classical result in the theory of polynomials is Descartes' Rule of Signs, which bounds the number of positive roots of a polynomial in terms of the number of sign changes in its coefficients. We use our extension of the Hermite-Biehler Theorem to give an inverse rule of signs for polynomials with one positive root.

Motivation & Objective

• To generalize the Hermite-Biehler Theorem beyond Hurwitz-stable polynomials by relating the number of real roots of $ p(-x^2) $ and $ q(-x^2) $ to the distribution of roots of $ f $ in the complex plane.
• To establish that the real roots of $ p(-x^2) $ and $ q(-x^2) $ interlace when $ f $ has $ n_- $ roots in the left half-plane and $ n_+ $ in the right, with the bound depending on $ |n_- - n_+| $.
• To provide an inverse rule of signs for real polynomials with exactly one positive root, showing that their even and odd coefficient sequences each have at most one sign change.
• To apply these results to problems in real root isolation and the Nonnegative Inverse Eigenvalue Problem (NIEP), particularly in characterizing coefficient sign patterns of polynomials with one positive root.
• To characterize the conditions under which consecutive coefficients vanish, identifying critical parameter thresholds in coefficient sequences.

Proposed method

• Decompose a real polynomial $ f(x) = p(x^2) + xq(x^2) $ into even and odd parts, analyzing the roots of $ p(-x^2) $ and $ q(-x^2) $.
• Use the argument principle and Rouché's theorem to relate the number of roots of $ f $ in the left and right half-planes to the number of real roots of $ p(-x^2) $ and $ q(-x^2) $.
• Establish interlacing of real roots of $ p(-x^2) $ and $ q(-x^2) $ via sign variation and root location analysis, generalizing the classical Hermite-Biehler condition.
• Apply inequalities from [4] (related to Newton’s inequalities) to analyze coefficient sign patterns, particularly under the assumption that all non-positive roots have nonpositive real parts.
• Use the factorization $ f(x) = (x - r)g(x) $ with $ r > 0 $, $ \operatorname{Re}(x_j) \leq 0 $ for $ j \geq 2 $, to derive recursive relations for coefficients $ a_k $ in terms of elementary symmetric functions of the roots.
• Prove that if $ a_k \leq 0 $, then $ a_{k+2} < 0 $, unless $ g(x) $ is of a specific form (e.g., reciprocal or purely imaginary), using equality conditions in coefficient inequalities.

Experimental results

Research questions

• RQ1 How can the Hermite-Biehler Theorem be extended to polynomials that are not Hurwitz stable, particularly in terms of the number and interlacing of real roots of $ p(-x^2) $ and $ q(-x^2) $?
• RQ2 What is the relationship between the number of roots of $ f $ in the open left and right half-planes and the number of real roots of $ p(-x^2) $ and $ q(-x^2) $?
• RQ3 Can an inverse rule of signs be established for polynomials with exactly one positive root, and what constraints does this impose on the sign patterns of their coefficients?
• RQ4 Under what conditions can consecutive coefficients of a polynomial with one positive root vanish, and how does this affect the sign variation in the coefficient sequence?
• RQ5 How do the coefficient sign patterns of such polynomials relate to the location of their non-positive roots in the complex plane, particularly in relation to the wedge $ S_{\sqrt{3}} $?

Key findings

• For a real polynomial $ f(x) = p(x^2) + xq(x^2) $ with $ n_- $ roots in the open left half-plane and $ n_+ $ in the open right half-plane, the polynomials $ p(-x^2) $ and $ q(-x^2) $ each have at least $ |n_- - n_+| $ real roots.
• The real roots of $ p(-x^2) $ and $ q(-x^2) $ interlace, even when $ f $ is not Hurwitz stable.
• If a real polynomial has exactly one positive root and all other roots have nonpositive real parts, then its even and odd coefficient sequences each contain at most one sign change.
• If $ f(x) = (x - r)g(x) $ with $ r > 0 $, $ \operatorname{Re}(x_j) \leq 0 $ for $ j \geq 2 $, and $ g $ not of the form $ \prod (x^2 + \beta_j^2) $, then $ a_k \leq 0 $ implies $ a_{k+2} < 0 $, ensuring at most one sign change in each coefficient sequence.
• The coefficient sequences $ \mathcal{T}_e $ and $ \mathcal{T}_o $ can each have at most one sign change unless $ f $ is of a special form (e.g., with purely imaginary roots or vanishing coefficients), and such cases are characterized via equality in coefficient inequalities.
• When $ \beta = \sqrt{2m+1} $ and $ r = 1 + 1/m $, the polynomial $ f(x) = (x - r)((x+1)^2 + \beta^2)^m $ has exactly one sign change, while increasing $ r $ beyond a threshold can lead to multiple sign changes, indicating a critical threshold in coefficient behavior.