Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Number of Real Types of Univariate Polynomials

Nicolas Faroß, Thomas Sturm

Abstract

The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the special case of a single polynomial we present a closed-form expression involving Fibonacci numbers. This allows us to precisely describe the asymptotic growth of the number of real types as the degree increases, in terms of the golden ratio.

On the Number of Real Types of Univariate Polynomials

Abstract

The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the special case of a single polynomial we present a closed-form expression involving Fibonacci numbers. This allows us to precisely describe the asymptotic growth of the number of real types as the degree increases, in terms of the golden ratio.

Paper Structure

This paper contains 10 sections, 20 theorems, 56 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Real types of single polynomials
  4. The structure of real types
  5. Real types of families of polynomials
  6. Counting real types of single polynomials
  7. Counting real types of families of polynomials
  8. Sign matrices depending on degrees
  9. Sign matrices independent of degrees
  10. Concluding remarks

Key Result

Proposition 3

∎ The number $R_d^{(m)}$ of real $d$-types realized by polynomials with exactly $m$ distinct real roots is given by

Figures (2)

  • Figure 1: The polynomial $f =x^4 - 4x^2$ and the family $f_1 = x + 1$, $f_2 = 2x + 1$, $f_3 = x^2 - 1$ of polynomials discussed in the Introduction
  • Figure 2: Python code for the computations in Example \ref{['ex:hong']}

Theorems & Definitions (40)

  • Example 1
  • Example 2
  • Proposition 3: Košta, 2016
  • Lemma 4: Košta, 2016; Lemma 4
  • Lemma 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • ...and 30 more