[Paper Review] Hyperbolic polynomials and rigid moduli orders

Vladimir Petrov Kostov|arXiv (Cornell University)|Aug 26, 2020

Mathematics and Applications10 references4 citations

TL;DR

This paper characterizes rigid moduli orders for hyperbolic polynomials—real univariate polynomials with all real roots—by determining when the relative ordering of root moduli (positive and negative) uniquely determines the sign pattern of the coefficients. It proves that only four specific interlacing orders (r_{PN}, r_{NP}, r_{PP}, r_{NN}) yield rigid moduli orders, each corresponding to one of two universal sign patterns: (+,+,-,-,+,+,...) or (+,-,-,+,+,-,...), depending on degree modulo 4.

ABSTRACT

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a 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 consider HPs with distinct moduli of the roots. We ask the question when the order of the moduli of the negative roots w.r.t. the positive roots on the real positive half-line completely determines the signs of the coefficients of the polynomial. When there is at least one positive and at least one negative root this is possible exactly when the moduli of the negative roots interlace with the positive roots (hence half or about half of the roots are positive). In this case the signs of the coefficients of the HP are either $(+,+,-,-,+,+,-,-,\ldots )$ or $(+,-,-,+,+,-,-,+,\ldots )$.

Motivation & Objective

• To determine which moduli orders of hyperbolic polynomials uniquely determine the sign pattern of their coefficients.
• To characterize rigid moduli orders—those that realize only one sign pattern—under the condition that all coefficients are nonvanishing.
• To establish a complete classification of such rigid orders and their associated sign patterns for hyperbolic polynomials with distinct root moduli.
• To extend the analysis to cases where some root moduli are equal, particularly when roots come in symmetric pairs ±a.
• To prove that only four specific interlacing patterns (r_{PN}, r_{NP}, r_{PP}, r_{NN}) are rigid, and that all others are not.

Proposed method

• Define a moduli order (MO) as a sequence of P (positive root) and N (negative root) symbols ordered by increasing modulus on the positive real line.
• Define a sign pattern (SP) as the sequence of signs of the polynomial’s coefficients, starting with +1 for the leading coefficient.
• Use Descartes’ rule of signs to relate the number of sign changes (c) and sign preservations (p) in the SP to the number of positive and negative roots.
• Apply inductive reasoning on the degree d, using perturbation techniques: construct one-parameter families of polynomials (Z_t) by deforming roots to analyze coefficient sign continuity.
• Use the fact that multiplying by (x±ε) or (1±εx) shifts the modulus of a new root to be smaller or larger than existing ones, preserving the MO structure.
• Analyze the coefficient signs of the product (x²−a²)Q(x) when roots are symmetric, showing that vanishing coefficients only occur in even degrees when Q is a product of quadratic factors.

Experimental results

Research questions

• RQ1 Which moduli orders of hyperbolic polynomials with distinct root moduli uniquely determine the sign pattern of their coefficients?
• RQ2 Under what conditions does a given moduli order fail to be rigid, i.e., realize more than one sign pattern?
• RQ3 What are the necessary and sufficient conditions for a moduli order to be rigid, particularly when the number of positive and negative roots are balanced?
• RQ4 How do coefficient sign patterns behave when symmetric root pairs (±a) are introduced, especially in even-degree polynomials?
• RQ5 Can the sign pattern of a hyperbolic polynomial be fully determined by the interlacing structure of the moduli of its positive and negative roots?

Key findings

• Only four specific moduli orders—r_{PN}, r_{NP}, r_{PP}, and r_{NN}—are rigid, meaning they uniquely determine the sign pattern of the coefficients.
• For these four rigid orders, the sign pattern is always one of two universal patterns: Σ₊ = (+,+,-,-,+,+,-,-,...) or Σ₋ = (+,-,-,+,+,-,-,+,...), depending on the degree modulo 4.
• The correspondence between the moduli order and the sign pattern is fully determined by the degree d modulo 4, as shown in the provided table.
• For non-rigid moduli orders (those not in {r_{PN}, r_{NP}, r_{PP}, r_{NN}}), multiple sign patterns can be realized, so the sign pattern is not uniquely determined.
• When a hyperbolic polynomial has a root pair ±a (i.e., symmetric roots), the coefficient signs depend on whether the remaining polynomial Q has vanishing coefficients; if d is odd, no coefficients vanish; if d is even, vanishing coefficients occur only if Q is a product of quadratic factors.
• In all cases with symmetric roots, the resulting sign pattern is either the universal Σ₊ or Σ₋ pattern, or the alternating pattern ( +,0,-,0,+,0,...) when d is even and Q is a product of (x²−a_j²) factors.