论文标题
数字半群的布奇威兹集
The Buchweitz set of a numerical semigroup
论文作者
论文摘要
令$ a \ subset {\ mathbb z} $为有限子集。我们用$ \ Mathcal {b}（a）$ $整数的集合$ n \ ge 2 $，这样$ | na | >（2n-1）（| a | -1）$，其中$ na = a+\ cdots+a $表示$ n $ fold $ a $ a $。考虑$ \ MATHCAL {B}（A）$的动机来自1980年Buchweitz的发现，如果数值半群$ S \ subseteq {\ Mathbb n} $是Weierstrass Semigroup，则是$ \ m nathcal {b}（b}（b}（\ Mathbb n} \ set $ setmins s）通过构建这种情况失败的实例，布赫威茨（Buchweitz）反驳了Hurwitz（1893）的长期猜想。在本文中，我们证明，对于任何数字半群$ s \ subset {\ mathbb n} $属的$ g \ ge 2 $，set $ \ mathcal {b}（{\ mathbb n} \ setminus s）$是有限的，是有限的，是$ s $ s $ $ s $。
Let $A \subset {\mathbb Z}$ be a finite subset. We denote by $\mathcal{B}(A)$ the set of all integers $n \ge 2$ such that $|nA| > (2n-1)(|A|-1)$, where $nA=A+\cdots+A$ denotes the $n$-fold sumset of $A$. The motivation to consider $\mathcal{B}(A)$ stems from Buchweitz's discovery in 1980 that if a numerical semigroup $S \subseteq {\mathbb N}$ is a Weierstrass semigroup, then $\mathcal{B}({\mathbb N} \setminus S) = \emptyset$. By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (1893). In this paper, we prove that for any numerical semigroup $S \subset {\mathbb N}$ of genus $g \ge 2$, the set $\mathcal{B}({\mathbb N} \setminus S) $ is finite, of unbounded cardinality as $S$ varies.
