Can a minimal degree 6 cubature rule for the disk have all points inside?

Authors:
Chirakkal Easwaran;Lawrence Fialkow;Srdjan Petrovic
Affiliations:
Department of Computer Science, State University of New York, New Paltz, NY 12561, USA;Department of Computer Science, State University of New York, New Paltz, NY 12561, USA;Department of Mathematics, Western Michigan University, Kalamazoo, MI 490085248, USA
Venue:
Journal of Computational and Applied Mathematics
Year:
2006

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Abstract

We use positivity and extension properties of moment matrices to prove that a 10-node (minimal) cubature rule of degree 6 for planar measure on the closed unit disk D@? cannot have all nodes in D@?. We construct examples showing that such rules may have as many as 9 points in D@?, and we provide similar examples for the triangle.