The Integer Sequence Transform $a \mapsto b$ where $b_n$ is the Number of Real Roots of the Polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$

TL;DR

This paper investigates the integer sequence transform mapping sequences to the number of real roots of associated polynomials, revealing sequences with trivial or interesting root patterns and methods to construct sequences with extremal root counts.

Contribution

It introduces the transform, characterizes sequences with trivial root counts, and provides procedures for constructing sequences with extremal real root numbers.

Findings

Sequences like Catalan numbers and factorials produce trivial root patterns.

Certain polynomial sequences yield more complex root behaviors.

Methods for constructing sequences with maximal or minimal real roots are presented.

Abstract

We discuss the integer sequence transform $a \mapsto b$ where $b_n$ is the number of real roots of the polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$. It is shown that several sequences $a$ give the trivial sequence $b = (0,1,0,1, 0,1,\ldots)$, i.e., ${b_n = n \bmod 2}$, among them the Catalan numbers, central binomial coefficients, $n!$ and $\binom{n+k}{n}$ for a fixed $k$. We also look at some sequences $a$ for which $b$ is more interesting such as $a_n = (n+1)^k$ for $k \geq 3$. Further, general procedures are given for constructing real sequences $a_n$ for which $b_n$ is either always maximal or minimal.