论文标题
高斯整数上最小欧几里得功能的基本证明
An Elementary Proof of the Minimal Euclidean Function on the Gaussian Integers
论文作者
论文摘要
每个欧几里得域$ r $具有最小的欧几里得功能，$ ϕ_r $。伴侣纸\ cite {graves}引入了一个公式来计算$ ϕ _ {\ mathbb {z} [i]} $。它是非平凡数字场整数环的最小欧几里得函数的第一个公式。通过研究集合的几何形状$ b_n = \ left \ {\ sum_ {j = 0} $ ϕ _ {\ mathbb {z} [i]}^{ - 1}（[0，n]）= b_n $提供$ ϕ _ {\ mathbb {z} [i]} $的简短证明。 Lenstra的证明需要实质性的代数背景。本文使用集合$ b_n $的新几何形状来证明$ ϕ _ {\ mathbb {z} [i]} $的公式，而无需使用lenstra的结果。新的几何方法使我们仅使用基本方法证明Lenstra的定理。然后，我们应用新公式来回答皮埃尔·塞缪尔（Pierre Samuel）的开放问题：$ ϕ _ {\ mathbb {z} [i]}^{ - 1}（n）$的大小是多少。附录提供了答案表和关联的SAGE代码。
Every Euclidean domain $R$ has a minimal Euclidean function, $ϕ_R$. A companion paper \cite{Graves} introduced a formula to compute $ϕ_{\mathbb{Z}[i]}$. It is the first formula for a minimal Euclidean function for the ring of integers of a non-trivial number field. It did so by studying the geometry of the set $B_n = \left \{ \sum_{j=0}^n v_j (1+i)^j : v_j \in \{0, \pm 1, \pm i \} \right \}$ and then applied Lenstra's result that $ϕ_{\mathbb{Z}[i]}^{-1}([0,n]) = B_n$ to provide a short proof of $ϕ_{\mathbb{Z}[i]}$. Lenstra's proof requires s substantial algebra background. This paper uses the new geometry of the sets $B_n$ to prove the formula for $ϕ_{\mathbb{Z}[i]}$ without using Lenstra's result. The new geometric method lets us prove Lenstra's theorem using only elementary methods. We then apply the new formula to answer Pierre Samuel's open question: what is the size of $ϕ_{\mathbb{Z}[i]}^{-1}(n)$?. Appendices provide a table of answers and the associated SAGE code.
