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

Abstract

The polynomial $f_{2n}(x)=1+x+\cdots +x^{2n}$ and its minimizer on the real line $x_{2n}=\mathrm{arginf}{f}_{2n}(x)$ for $n\in \mathbb{N}$ 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.

12 pages, 2 figures