On the Number of Real Types of Univariate Polynomials

TL;DR

This paper provides explicit formulas for counting the distinct sign behaviors of families of univariate polynomials over the real line, revealing asymptotic growth patterns related to Fibonacci numbers and the golden ratio.

Contribution

It introduces a formula for the number of real types of polynomial families with degree bounds and a closed-form expression for single polynomials involving Fibonacci numbers.

Findings

Number of real types grows exponentially with degree.

Explicit formulas involve Fibonacci numbers and the golden ratio.

Asymptotic growth rate is characterized precisely.

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.