On the uniqueness of algebraic curves passing through n-independent nodes

摘要

A set of nodes in the plane is called n-independent if for arbitrary data atthose nodes, there is a (not necessarily unique) polynomial of degree at most n that matches the giveninformation. We proved in a previous paper (Hakopian-Toroyan, 2015) that theminimal number of n-independent nodes determining uniquely the curveof degree k≦ n passing through them equals to D:=(1/2)(k-1)(2n+4-k)+2. In this paper we bring a characterization of the case when at leasttwo curves of degree k pass through the nodes of an n-independentnode set of cardinality D-1. Namely, we prove that the latter set has a very special construction: All its nodes but one belong to a (maximal) curve of degree k-1.We show that this result readily yields the above cited one. At the end, an important application to the Gasca-Maeztu conjecture is presented.