On a Scale of Criteria on n-Dependence

摘要

In this paper we prove that a planar set X of at most mn - 1 points, where m = n, is kappa-dependent if and only if there exists a number r, 1 = r = m - 1, and an essentially kappa-dependent subset y subset of X, #y = 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 #y = rs, then the set 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.