On the area bounded by the curve $\prod_{k = 1}^n\left|x\sin\frac{k\pi}{n} - y\cos\frac{k\pi}{n}\right|=1$

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 $4^{1-1∕n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$, 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

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

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 $n$.