论文标题
阿贝尔组中序列的总和和字段扩展中的标志
Sumsets of sequences in abelian groups and flags in field extensions
论文作者
论文摘要
对于有限的Abelian Group $ g $带子集$ a $和$ b $，集中$ ab $ is $ \ {ab \ {ab \ mid a \ in A，b \ in B \} $。添加剂组合学的一个基本问题是，就$ a $ a $ a $ a $ a $ a $ a $ ab $的基数的基数下降。本文解决了阿贝尔组和场扩展中标志的序列的类似问题。对于一个正整数$ n $，令$ [n] $表示集合$ \ {0，\ dots，n-1 \} $。到有限的Abelian组$ g $ of Cardinality $ n $和一个订购$ g = \ {1 = v_0，\ dots，v_ {n-1} \} $，将功能$ t \ colon [n] \ times [n] \ rightArrow [n] \ rightArrow [n] $与[ t（i，j）= \ min \ big \ {k \ in [n] \ mid \ {v_0，\ dots，v_i \} \ {v_0，\ dots，\ dots，v_j \} \ v_j \} \ subseteq \ subseteq \ subseteq \ subseteq \ subseteq \ {v_0，\ dots，\ dots，\ dots，\ dots，v_k \ \ v_k \ \} \]在自然部分订购下，哪些功能$ t $最小为$ \ {1 = v_0，\ dots，v_ {n-1} \} $范围，范围是有限的Abelian Cardinality $ n $的订购的订单？我们还向类似的问题提出了$ n $ field扩展的程度。 当$ n <18 $，$ n $是主要功率，或$ n $是$ 2 $不同的素数时，我们将明确对所有最小$ t $进行分类。当$ n $不那么上述时，我们明确地构建了上述分类中不包含关联功能$ t $的阿贝尔集团的顺序。我们还将订购的多面体编码编码$ t $的数据。
For a finite abelian group $G$ with subsets $A$ and $B$, the sumset $AB$ is $\{ab \mid a\in A, b \in B\}$. A fundamental problem in additive combinatorics is to find a lower bound for the cardinality of $AB$ in terms of the cardinalities of $A$ and $B$. This article addresses the analogous problem for sequences in abelian groups and flags in field extensions. For a positive integer $n$, let $[n]$ denote the set $\{0,\dots,n-1\}$. To a finite abelian group $G$ of cardinality $n$ and an ordering $G = \{1=v_0,\dots,v_{n-1}\}$, associate the function $T \colon [n] \times [n] \rightarrow [n]$ defined by \[ T(i,j) = \min\big\{k \in [n] \mid \{v_0,\dots,v_i\}\{v_0,\dots,v_j\} \subseteq \{v_0,\dots,v_k\}\big\}. \] Under the natural partial ordering, what functions $T$ are minimal as $\{1=v_0,\dots,v_{n-1}\}$ ranges across orderings of finite abelian groups of cardinality $n$? We also ask the analogous question for degree $n$ field extensions. We explicitly classify all minimal $T$ when $n < 18$, $n$ is a prime power, or $n$ is a product of $2$ distinct primes. When $n$ is not as above, we explicitly construct orderings of abelian groups whose associated function $T$ is not contained in the above classification. We also associate to orderings a polyhedron encoding the data of $T$.
