Mathematics – Rings and Algebras
Scientific paper
2010-04-08
Adv. Math. 226 (2011), no. 2, 1202-1212
Mathematics
Rings and Algebras
10 pages. To appear in Advances in Mathematics
Scientific paper
10.1016/j.aim.2010.08.003
There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any real zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a real zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.
No associations
LandOfFree
Obstructions to determinantal representability does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Obstructions to determinantal representability, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Obstructions to determinantal representability will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-713465