تقرير
Exact and approximate solutions to the minimum of $1+x+\cdots+x^{2n}$
| العنوان: | Exact and approximate solutions to the minimum of $1+x+\cdots+x^{2n}$ |
|---|---|
| المؤلفون: | Hendrickson, Aaron, Leibovici, Claude F. |
| سنة النشر: | 2021 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - History and Overview, Mathematics - Classical Analysis and ODEs, 65H04 (Primary) 65Q30 (Secondary) |
| الوصف: | 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. Comment: 12 pages, 2 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2105.00135 |
| رقم الأكسشن: | edsarx.2105.00135 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |