The sets of nodes in the plane for which its nth degree Lagrange polynomials can be factored as a product of first degree polynomials satisfy a geometric characterization: for each node there exists a set of n lines containing the other nodes. Generalized principal lattices are sets of nodes defined by three families of lines. A generalized principal lattice satisfies the geometric characterization and there exist exactly three lines in the plane containing more nodes than the degree. In this paper, we show a converse, valid for degrees n 7: if a set of nodes satisfy the geometric characterization and there exist exactly three lines containing n + 1 nodes, then it is a generalized principal lattice.
Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4.
The geometric characterization identifies the sets of nodes such that the Lagrange polynomials are products of factors of first degree. We offer a detailed classification of all known sets satisfying the geometric characterization in the plane. The defect, which takes into account the number of lines containing more nodes than the degree, plays a fundamental role in this classification.
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