论文标题
关于$ gc_n $ sets的基本属性
On the basic properties of $GC_n$ sets
论文作者
论文摘要
平面节点集$ \ mathcal x，$带有$ \＃\ Mathcal x = \ binom {n+2} {2} {2}，如果每个节点都具有$ n $线性因子的产物形式的基本多项式，则称为$ gc_n $ set。我们说，如果线是节点的基本多项式的一个因素，则使用线路。一条线称为$ k $ node行，如果它通过$ \ natercal x的$ k $ nodes通过$ \ mathcalX。$最多可以在任何$ n+1 $ nodes中插入任何$ gc_n $ set中，而$（n+1）$ - 节点线称为最大线。 Gasca-Maeztu猜想（1982）指出，每$ gc_n $ set都有最大线。到目前为止，猜想仅证明了$ n \ le 5的案例。 在这里，对于一行$ \ ell $，我们介绍并研究了$ \ ell $ - 换$ \ Mathcal x $的概念，并定义所谓的正确行。我们还提供了$ GC_N $集的几种基本属性，内容涉及最大线，$ n $ - 节点线，用过的线路以及使用给定线路的节点的子集。
A planar node set $\mathcal X,$ with $\#\mathcal X=\binom{n+2}{2},$ is called $GC_n$ set if each node possesses fundamental polynomial in form of a product of $n$ linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of the node. A line is called $k$-node line if it passes through exactly $k$-nodes of $\mathcal X.$ At most $n+1$ nodes can be collinear in any $GC_n$ set and an $(n+1)$-node line is called a maximal line. The Gasca-Maeztu conjecture (1982) states that every $GC_n$ set has a maximal line. Until now the conjecture has been proved only for the cases $n \le 5.$ Here, for a line $\ell$ we introduce and study the concept of $\ell$-lowering of the set $\mathcal X$ and define so called proper lines. We also provide refinements of several basic properties of $GC_n$ sets regarding the maximal lines, $n$-node lines, the used lines, as well as the subset of nodes that use a given line.
