论文标题
在曲线界限的区域上= 1 $
On the Area Bounded by the Curve $\prod_{k = 1}^n |x\sin(kπ/n)-y\cos(kπ/n)| = 1$
论文作者
论文摘要
对于正整数$ n $，让$$ f_n^*（x，y）= \ prod \ limits_ {k = 1}^n \ left（x \ sin \ left（\ frac {kπ} {kπ} {n} {n} \ right）-y \ cos \ cos \ cos \ weft（\ frac {kπ} n} n} n} {n} {n} {n} { 2000年Bean和Laugesen中的$$证明，每$ n \ geq 3 $ curve $ | f_n^*（x，y）| = 1 $等于$ 4^{1-1/n} b \ left（\ frac {1} {2} {2} - \ frac {1} {n}，\ frac {1} {2} {2} {2} \ right）$，其中$ b（x，y）$是beta函数。我们根据面积计算的极性公式提供了这一事实的基本证明。我们还证明了$$ f_n^*（x，y）= 2^{1 -n} \ sum \ limits _ {\ ordack {1 \ leq k \ leq k \ leq n \\\ text {$ k $ as od od}}}}}}}}}}}}}}}}}}}}}}}}}（ - 1） -k} y^k $$并证明$ \ ell_n = 2^{n -1 -ν_2（n）} $是最小的正整数，因此二进制形式$ \ ell_n f_n f_n^*（x，x，y）$具有integer系数。这里$ν_2（n）$表示$ 2 $ -ADIC订单的$ N $。
For a positive integer $n$, let $$ F_n^*(X, Y) = \prod\limits_{k = 1}^n\left(X\sin\left(\frac{kπ}{n}\right) -Y\cos\left(\frac{kπ}{n}\right)\right). $$ In 2000 Bean and Laugesen proved that for every $n \geq 3$ the area bounded by the curve $|F_n^*(x, y)| = 1$ 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 $$ F_n^*(X, Y) = 2^{1 - n}\sum\limits_{\substack{1 \leq k \leq n\\\text{$k$ is odd}}}(-1)^{\frac{k - 1}{2}}\binom{n}{k}X^{n - k}Y^k $$ and demonstrate that $\ell_n = 2^{n - 1 - ν_2(n)}$ is the smallest positive integer such that the binary form $\ell_n F_n^*(X, Y)$ has integer coefficients. Here $ν_2(n)$ denotes the $2$-adic order of $n$.
