## Abstract

For a positive integer $n$, let

$${F}_{n}^{\ast}\left(X,Y\right)=\prod _{k=1}^{n}\left(Xsin\frac{k\pi}{n}-Ycos\frac{k\pi}{n}\right).$$

In 2000 Bean and Laugesen proved that for every $n\ge 3$ the area bounded by the curve $\left|{F}_{n}^{\ast}\left(x,y\right)\right|=1$ is equal to ${4}^{1-1\u2215n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$, where $B\left(x,y\right)$ is the beta function. We provide an elementary proof of this fact based on the polar formula for the area calculation. We also prove that

$${F}_{n}^{\ast}\left(X,Y\right)={2}^{1-n}Y\sum _{\begin{array}{c}1\le k\le n\\ \text{}k\text{isodd}\end{array}}{\left(-1\right)}^{\frac{k-1}{2}}\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){X}^{n-k}{Y}^{k}$$

and demonstrate that ${\ell}_{n}={2}^{n-1-{\nu}_{2}\left(n\right)}$ is the smallest positive integer such that the binary form ${\ell}_{n}{F}_{n}^{\ast}\left(X,Y\right)$ has integer coefficients. Here ${\nu}_{2}\left(n\right)$ denotes the $2$-adic order of $n$.

## Citation

Anton Mosunov. "On the area bounded by the curve $\prod_{k = 1}^n\left|x\sin\frac{k\pi}{n} - y\cos\frac{k\pi}{n}\right|=1$." Rocky Mountain J. Math. 50 (5) 1773 - 1777, October 2020. https://doi.org/10.1216/rmj.2020.50.1773