On a scale of criteria on $n$-dependence

Abstract

In this paper we prove that a planar set $\mathcal{X}$ of at most $mn-1$ points, where $m \le n$, is $\kappa$-dependent, if and only if there exists a number r, $1 \le r \le m-1$, and an essentially $\kappa$-dependent subset $\mathcal{Y} \subset \mathcal{X}$, $\#\mathcal{Y} \ge rs$, 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.