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.<p>Favorite paper: minimal tetrahedralization of a <i>convex</i> polyhedron is NP-hard.
<a href="http://arxiv.org/PS_cache/math/pdf/0012/0012177v1.pdf" rel="nofollow">http://arxiv.org/PS_cache/math/pdf/0012/0012177v1.pdf</a><p><a href="http://www.math.ucdavis.edu/~deloera/RECENT_WORK/recent.html" rel="nofollow">http://www.math.ucdavis.edu/~deloera/RECENT_WORK/recent.html</a>