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

## 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 $|{F}_{n}^{\ast }\left(x,y\right)|=1$ 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

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

## Information

Received: 1 February 2020; Accepted: 9 February 2020; Published: October 2020
First available in Project Euclid: 5 November 2020

zbMATH: 07274833
MathSciNet: MR4170685
Digital Object Identifier: 10.1216/rmj.2020.50.1773

Subjects:
Primary: 11D75, 51M25
Secondary: 11J25, 33B15  