Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

In search of Newton-type inequalities

Olga Katkova, Boris Shapiro, Anna Vishnyakova

Abstract

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide counterexamples to two earlier conjectures refining Descartes rule of signs.

In search of Newton-type inequalities

Abstract

In this paper, we prove a number of results providing either necessary or sufficient conditions guaranteeing that the number of real roots of real polynomials of a given degree is either less or greater than a given number. We also provide counterexamples to two earlier conjectures refining Descartes rule of signs.
Paper Structure (8 sections, 16 theorems, 194 equations)

This paper contains 8 sections, 16 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Main results
  4. Previously known facts
  5. New results
  6. Proofs
  7. Related conjectures and some counterexamples
  8. Final remark and outlook

Key Result

Proposition 1

(i) The polyhedral cone described by (the logarithm of) Hutchinson’s inequalities eq:Hu is the maximal polyhedral cone contained in $\text{Log}\,(\Sigma_n^+)$. (ii) The minimal polyhedral cone containing $\text{Log}\,(\Sigma_n^+)$ is given by (the logarithm of) Newton’s inequalities.

Theorems & Definitions (30)

  • Remark 1
  • Proposition 1: see Proposition 7 of KoSh
  • Remark 2
  • Definition 1
  • Definition 2
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Remark 3
  • ...and 20 more