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On Descartes' rule of signs for hyperbolic polynomials

Vladimir Petrov Kostov

Abstract

We consider univariate real polynomials with all roots real and with two sign changes in the sequence of their coefficients which are all non-vanishing. One of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree $d$ polynomials have $2$ positive and $d-2$ negative roots. We consider the sequences of the moduli of their roots on the real positive half-axis. When the moduli are distinct, we give the exhaustive answer to the question at which positions can the moduli of the two positive roots be.

On Descartes' rule of signs for hyperbolic polynomials

Abstract

We consider univariate real polynomials with all roots real and with two sign changes in the sequence of their coefficients which are all non-vanishing. One of the changes is between the linear and the constant term. By Descartes' rule of signs, such degree polynomials have positive and negative roots. We consider the sequences of the moduli of their roots on the real positive half-axis. When the moduli are distinct, we give the exhaustive answer to the question at which positions can the moduli of the two positive roots be.
Paper Structure (5 sections, 10 theorems, 62 equations)

This paper contains 5 sections, 10 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Proof of part (A) of Theorem \ref{['tmresult']}
  3. Proof of part (B) of Theorem \ref{['tmresult']}
  4. Proof of part (C) of Theorem \ref{['tmresult']}
  5. Proofs of Lemmas \ref{['lmn-1n']}, \ref{['lm2n1']} and \ref{['lm351']}

Key Result

Theorem 1

The sign pattern $\Sigma _{m,n}$, $1\leq n\leq m$, is realizable with and only with orders such that $\alpha _1<\gamma _{2n-1}$. For $1\leq m\leq n$, it is realizable with and only with orders such that $\gamma _{d-2m}<\alpha _1$.

Theorems & Definitions (23)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof : Proof of Theorem \ref{['tm1']}
  • ...and 13 more