Exact and approximate solutions to the minimum of $1+x+\cdots+x^{2n}$

TL;DR

This paper investigates the exact and approximate solutions for the minimum point of the polynomial sum $1+x+ o+x^{2n}$, providing theoretical insights and practical approximation methods.

Contribution

It introduces a precise characterization of the minimizer, derives an exact hypergeometric sum expression, and develops perturbation-based approximations for the polynomial's minimum.

Findings

The minimizer exists, is unique, and lies in [-1, -1/2].

Exact solution for the minimizer involves hypergeometric functions.

Perturbation theory yields rapidly converging approximations.

Abstract

The polynomial $f_{2n}(x)=1+x+\cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=\operatorname{arg\,inf} f_{2n}(x)$ for $n\in\Bbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $\partial_x f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is further shown that $\inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $\inf f_{2n}(x)\in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in the form of a finite sum of hypergeometric functions of unity argument. Perturbation theory is applied to generate rapidly converging and asymptotically exact approximations to $x_{2n}$. Numerical studies are carried out to show how many terms of the perturbation expansion for $x_{2n}$ are needed to obtain suitably accurate approximations to the exact value.