I know this post is about Parrilo but I would just like to point out that Jesus De Loera (co-author of linked paper) is awesome.

Favorite paper: minimal tetrahedralization of a convex polyhedron is NP-hard. http://arxiv.org/PS_cache/math/pdf/0012/0012177v1.pdf

http://www.math.ucdavis.edu/~deloera/RECENT_WORK/recent.html