论文标题
反环体多项式的正交关系
An orthogonal relation on inverse cyclotomic polynomials
论文作者
论文摘要
令$φ_n（x）$和$ψ_n（x）= \ frac {x^{n} -1} {φ_{n}（x）（x）} $是$ n $ - th cyclotomic和逆环环体元素。在简短的说明中，对于任何一对除数$ d_ {1} \ neq d_ {2} $的$ n $，以及integers $ l_1 $和$ l_2 $，以至于$ 0 \ leq l_ {1}} \ leqleqφ（d_ {1}我们表明\ [\ left \ langle x^{l_ {1}}ψ_{d_ {d_ {1}}}（x）（1+x^{d_1}+\ dots x^{n-d_1}） （1+x^{d_2}+\ dots x^{n-d_2}）\ right \ rangle = 0，\]其中$ \ langle \ cdot，\ cdot \ rangle $是$ \ mathbb {q} [q} [q} [x] $ wardy $ \ langle \ langle \ sum__ _ sum_ \ sum_ _}的内部产品a_ {k} x^{k}，\ sum_ {k} b_ {k} x^{k} \ rangle = \ sum_ {k} a_ {k} b_ {k {k {k} $。
Let $Φ_n(X)$ and $Ψ_n(X)=\frac{X^{n}-1}{Φ_{n}(X)}$ be the $n$-th cyclotomic and inverse cyclotomic polynomials respectively. In this short note, for any pair of divisors $ d_{1} \neq d_{2} $ of $ n $, and integers $l_1$ and $l_2$ such that $ 0 \leq l_{1} \leq φ(d_{1})-1 $ and $ 0 \leq l_{2} \leq φ(d_{2})-1 $, we show that \[\left \langle X^{l_{1}} Ψ_{d_{1}}(X) (1+X^{d_1}+\dots X^{n-d_1}), X^{l_{2}} Ψ_{d_{2}}(X) (1+X^{d_2}+\dots X^{n-d_2}) \right \rangle =0, \] where $ \langle \cdot, \cdot \rangle $ is the inner product on $\mathbb{Q}[X]$ defined by $ \langle \sum_{k} a_{k}X^{k},\sum_{k} b_{k}X^{k} \rangle =\sum_{k} a_{k}b_{k}$.
