Abstract

In this paper we prove that a planar set \(\mathcal{X}\) of at most \(mn-1\) points, where \(m\leq n\), is \(\kappa\)-dependent if and only if there exists a number \(r\), \(1\leq r\leq m-1\), and an essentially \(\kappa\)-dependent subset \(\mathcal{Y}\subset\mathcal{X}\), \(\#\mathcal{Y}\geq rs\), where \(r+s-3=\kappa\), 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.

中文翻译：

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在$$ \ boldsymbol {n} $$的标准尺度上-依赖

摘要

在本文中我们证明了的平面集合\（\ mathcal {X} \）至多的\（MN-1 \）点，其中\（米\当量Ñ\） ，是\（\卡帕\）依赖性如果并且仅当存在数字\（r \），\（1 \ leq r \ leq m-1 \）和本质上依赖于（（kappa \）的子集\（\ mathcal {Y} \ subset \ mathcal {X} \），\（\＃\ mathcal {Y} \ geq rs \），其中\（r + s-3 = \ kappa \）属于度数\（r \）的代数曲线，而不是属于任何小于\（r \）的度数曲线。此外，如果\（\＃\ mathcal {Y} = rs \），则集合\（\ mathcal {Y} \）分别与度数\（r \）和\（s \）的某些两条曲线的交点集重合。让我们提到，对于\（m = 1,2,3，\），量表的前三个标准是众所周知的结果。