On the Area Bounded by the Curve ${\prod }_{k=1}^n|x\sin (k\pi /n)-y\cos (k\pi /n)|=1$

Abstract

For a positive integer $n$, let $$F_n^{\ast }(X,Y)=\prod _{k=1}^n\left(X\sin \left(\frac{k\pi }{n}\right)-Y\cos \left(\frac{k\pi }{n}\right)\right).$$ In 2000 Bean and Laugesen proved that for every $n\ge 3$ the area bounded by the curve $|F_n^{\ast }(x,y)|=1$ is equal to $4^{1-1/n}B\left(\frac{1}{2}-\frac{1}{n},\frac{1}{2}\right)$, where $B(x,y)$ 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 $$F_n^{\ast }(X,Y)=2^{1-n}\sum _{\begin{array}{c}1\le k\le n\\ \text{k is odd}\end{array}}(-1{)}^{\frac{k-1}{2}}(\genfrac{}{}{0ex}{}{n}{k})X^{n-k}{Y}^k$$ and demonstrate that ${\ell }_n=2^{n-1-{\nu }_2(n)}$ is the smallest positive integer such that the binary form ${\ell }_n{F}_n^{\ast }(X,Y)$ has integer coefficients. Here ${\nu }_2(n)$ denotes the $2$-adic order of $n$.

7 pages