论文标题
完全真实数字字段中完美的单一形式的类数量的上限
An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields
论文作者
论文摘要
令$ k $为$ \ m athbb {q} $的$ n $的完全实数字段，分别具有判别和调节器$Δ_K，r_k $。在本文中，使用与van Woerden类似的方法，我们证明，完美的一般形式的类别（最多与等效性和缩放）可以通过$ o（δ_k\ exp（2n \ log log（n）+f（n）+f（n，r_k））$，其中$ f（n，r_k）$是有限的，满足的值，满足$ f（n，r_k）= \ frac {\ sqrt {n-1}} {2} r_k^{\ frac {\ frac {1} {n-1}}}+\ frac {4} {n-1} {n-1} {n-1} {n-1} \ log（此外，如果$ k $是一个单位还原字段，那么完美的一单单表格的类数将在上面绑定为$ o（δ_k\ exp（2n \ log（n）））$。
Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $Δ_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by $O( Δ_K \exp(2n \log(n)+f(n,R_K)))$, where $f(n,R_K)$ is a finite value, satisfying $f(n,R_K)=\frac{\sqrt{n-1}}{2}R_K^{\frac{1}{n-1}}+\frac{4}{n-1}\log(\sqrt{|Δ_K|})^2$ if $n \leq 11$. Moreover, if $K$ is a unit reducible field, the number of classes of perfect unary forms is bound above by $O( Δ_K \exp(2n \log(n)))$.
