On the basic properties of GCn sets

Abstract

In this paper the simplest [Formula: see text]-correct sets in the plane — [Formula: see text] sets are studied. An [Formula: see text]-correct node set [Formula: see text] is called [Formula: see text] set if the fundamental polynomial of each node is a product of [Formula: see text] linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. A line is called [Formula: see text]-node line if it passes through exactly [Formula: see text] nodes of [Formula: see text] At most [Formula: see text] nodes can be collinear in any [Formula: see text]-correct set and an [Formula: see text]-node line is called a maximal line. The Gasca–Maeztu conjecture (1982) states that for every [Formula: see text] set there exists at least one maximal line. Until now the conjecture has been proved only for the cases [Formula: see text] Here, for a line [Formula: see text] we introduce and study the concept of [Formula: see text]-lowering of the set [Formula: see text] and define so called proper lines. We also provide refinements of several basic properties of [Formula: see text] sets regarding the maximal lines, [Formula: see text]-node lines, the used lines, as well as the subset of nodes that use a given line.