论文标题
环形整数的Cassels高度
The Cassels heights of cyclotomic integers
论文作者
论文摘要
我们研究了环形整数$β$的模量的均方根值的集合$ \ Mathscr c $。对于它的$ k $ th派生的集合$ \ mathscr c^{（k）} $，我们表明$ \ Mathscr c^{（k）} =（k+1）\ mathscr c \，\，（k \ ge 0）$ c}^{（\ ell）} = {\ Mathscr C}^{（K+\ ell+1）} \，\，（K，\ Ell \ ge 0）$。我们还计算$ \ Mathscr c $的订单类型，并表明它与PV数字集相同。 此外，我们精确地描述了限制的$ \ mathscr c_p $，其中$β$仅限于ring $ \ m rathbb z [ω_p] $，其中$ p $是奇数prime，$ω_p$是原始$ p $ p $ p $ p $ th root of Unity。为此，我们证明了两个二次多项式$ a^2+ab+b^2+c^2+a+a+b+c $和$ a^2+b^2+c^2+c^2+ab+ab+bc+ca+a+a+a+b+c $是通用的。
We study the set $\mathscr C$ of mean square values of the moduli of the conjugates of cyclotomic integers $β$. For its $k$th derived set $\mathscr C^{(k)}$, we show that $\mathscr C^{(k)}=(k+1)\mathscr C\,\, (k\ge 0)$, so that also ${\mathscr C}^{(k)}+{\mathscr C}^{(\ell)}={\mathscr C}^{(k+\ell+1)}\,\,(k,\ell\ge 0)$. We also calculate the order type of $\mathscr C$, and show that it is the same as that of the set of PV numbers. Furthermore, we describe precisely the restricted set $\mathscr C_p$ where the $β$ are confined to the ring $\mathbb Z[ω_p]$, where $p$ is an odd prime and $ω_p$ is a primitive $p$th root of unity. In order to do this, we prove that both of the quadratic polynomials $a^2+ab+b^2+c^2+a+b+c$ and $a^2+b^2+c^2+ab+bc+ca+a+b+c$ are universal.
