Journal
SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY
Volume 5, Issue 3, Pages 526-551Publisher
SIAM PUBLICATIONS
DOI: 10.1137/19M1289820
Keywords
phaseless rank; equimodular matrices; amoebas; semidefinite rank; polytopes
Categories
Funding
- Centre for Mathematics of the University of Coimbra - Portuguese Government through FCT/MEC [UID/MAT/00324/2019]
- European Regional Development Fund
- FCT [PD/BD/135276/2017]
- Fundação para a Ciência e a Tecnologia [PD/BD/135276/2017] Funding Source: FCT
Ask authors/readers for more resources
In this paper, the concept of phaseless rank of a complex matrix is discussed and connected to determinantal varieties, semidefinite representations of convex sets, and the complexity of polytopes. New results are derived and an upper bound on the complex semidefinite extension complexity of polytopes is obtained. The study also highlights the connections between phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases.
We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this quantity, extending a classic result of Camion and Hoffman and connecting it to the study of amoebas of determinantal varieties and of semidefinite representations of convex sets. As a consequence, we prove that the set of maximal minors of a matrix of indeterminates form an amoeba basis for the ideal they define, and we attain a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. We also highlight the connections between the notion of phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available