论文标题
根据$ n $依赖的标准规模
On a scale of criteria on $n$-dependence
论文作者
论文摘要
In this paper we prove that a planar set $\mathcal{X}$ of at most $mn-1$ points, where $m \le n$, is $κ$-dependent, if and only if there exists a number r, $1 \le r \le m-1$, and an essentially $κ$-dependent subset $\mathcal{Y} \subset \mathcal{X}$, $ \＃\ Mathcal {y} \ ge rs $，其中$ r + s -3 =κ$，属于$ r $的代数曲线，不属于任何小于$ r $的曲线。此外，如果$ \＃\ MATHCAL {y} = rs $，则设置$ \ Mathcal {y} $与大约两个学位$ r $和$ s $的两个曲线的相交点相吻合。 让我们提到的是，该量表的前三个标准，价格为$ m = 1,2,3，$是众所周知的结果。
In this paper we prove that a planar set $\mathcal{X}$ of at most $mn-1$ points, where $m \le n$, is $κ$-dependent, if and only if there exists a number r, $1 \le r \le m-1$, and an essentially $κ$-dependent subset $\mathcal{Y} \subset \mathcal{X}$, $\#\mathcal{Y} \ge rs$, where $r + s - 3 = κ$, belonging to an algebraic curve of degree $r$, and not belonging to any curve of degree less than $r$. Moreover, if $\#\mathcal{Y} = rs$ then the set $\mathcal{Y}$ coincides with the set of intersection points of some two curves of degrees $r$ and $s$, respectively. Let us mention that the first three criteria of the scale, for $m=1,2,3,$ are well-known results.
