2021/11/03 by António Pedro Goucha, Goucha, António Pedro, João Gouveia +1
Computer Science · Engineering · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2111.01962
openalex publication_date 2021/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this paper, we introduce and study the notion of phase rank. Given a matrix filled with phases, i.e., with complex entries of modulus 1, its phase rank is the smallest possible rank for a complex matrix with the same phase, but possible different modulus. This notion generalizes the notion of sign rank, and is the complementary notion to that of phaseless rank. It also has intimate connections to the problem of ray nonsingularity and provides an important class of examples for the study of coamoebas. Among other results, we provide a new characterization of phase rank 2 for 3 × 3 matrices, that implies the new result that for the 3× 3 determinantal variety, its coamoeba is determined solely by the colopsidedness criterion.