[Paper Review] Hyperbolic polynomials and canonical sign patterns

Vladimir Petrov Kostov|arXiv (Cornell University)|Jun 25, 2020

Mathematics and Applications10 references7 citations

TL;DR

This paper investigates the canonical order of moduli of roots in hyperbolic polynomials (HPs) with distinct positive root moduli, focusing on which sign patterns of coefficients force exactly one such order. Using a recursive concatenation method that preserves coefficient signs while controlling root modulus ordering, the authors characterize canonical sign patterns—those realizable by HPs only in the canonical root order. The key result identifies all such patterns via structural constraints on coefficient sign sequences and their correspondence to root ordering via change-preservation patterns.

ABSTRACT

A real univariate polynomial is hyperbolic if all its roots are real. By Descartes' rule of signs a hyperbolic polynomial (HP) with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. We discuss the question: If the moduli of all $c+p$ roots are distinct and ordered on the positive half-axis, then at which positions can the $p$ moduli of negative roots be depending on the positions of the positive and negative signs of the coefficients of the polynomial? We are especially interested in the choices of these signs for which exactly one order of the moduli of the roots is possible.

Motivation & Objective

• To determine which sign patterns of coefficients in hyperbolic polynomials force a unique ordering of the moduli of their roots on the positive real axis.
• To define and characterize 'canonical sign patterns'—those for which only the canonical root modulus order is realizable by a hyperbolic polynomial.
• To establish a constructive method for generating hyperbolic polynomials with prescribed sign patterns and canonical root modulus order.
• To clarify the relationship between coefficient sign sequences, Descartes' rule of signs, and the realizability of specific root orderings.

Proposed method

• Constructs hyperbolic polynomials recursively via concatenation: appending roots with moduli much smaller or larger than existing ones to preserve coefficient signs.
• Uses a bijective correspondence between sign patterns (SPs) and change-preservation patterns (CPPs), where 'c' denotes sign change and 'p' denotes sign preservation.
• Defines the canonical order of root moduli by reversing the CPP and replacing 'c' with 'P' (positive root) and 'p' with 'N' (negative root).
• Employs perturbation techniques to ensure distinct root moduli while preserving coefficient signs and achieving arbitrary root orderings within constraints.
• Applies deformation of known polynomials (e.g., $P_{\ell}$ with vanishing coefficients) to construct examples realizing specific sign patterns.
• Uses recursive concatenation with $x + \varepsilon$ (small $\varepsilon$) to add small-modulus negative roots and $1 + \varepsilon x$ to add large-modulus negative roots, preserving sign patterns.

Experimental results

Research questions

• RQ1 Which sign patterns of coefficients in a hyperbolic polynomial force exactly one possible order of the moduli of its roots on the positive real axis?
• RQ2 What structural properties of a sign pattern ensure that only the canonical root modulus order is realizable?
• RQ3 How can one systematically construct a hyperbolic polynomial with a given sign pattern and canonical root modulus order?
• RQ4 To what extent do Descartes’ rule of signs and the change-preservation pattern determine the realizability of a specific root ordering?

Key findings

• Every sign pattern of length $d+1$ beginning with '+' is realizable by a degree $d$ hyperbolic polynomial with the canonical order of root moduli.
• Canonical sign patterns are those for which only the canonical root modulus order is realizable, and they are characterized by specific structural constraints on the sequence of sign changes and preservations.
• For sign patterns of the form $\Sigma_{m_1,1,m_3}$ with $m_1, m_3 \geq 1$, the canonical order is uniquely realizable.
• When $\ell = r^2 - 3$, the polynomial $P_{\ell}$ has vanishing coefficients at positions $r(r-1)/2$ and $r(r+1)/2$, enabling construction of sign patterns with adjacent positive and negative root moduli in canonical order.
• The sign pattern $\sigma(P_+) = (r(r-1)/2 + 1, r, r(r-1)/2)$ and $\sigma(P_-) = (r(r-1)/2, r, r(r-1)/2 + 1)$ are realizable with canonical ordering, showing flexibility in canonical pattern construction.
• By concatenating with $x + \varepsilon_j$ (small $\varepsilon_j$) and $1 + \varepsilon_j x$ (small $\varepsilon_j$), one can extend any canonical sign pattern to include additional negative roots of arbitrarily small or large modulus, preserving the canonical order.