2020/11/10 by C P Anil Kumar, Kumar, C P Anil
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Polynomial and algebraic computation #Primary: 52C35 #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2011.05327
openalex publication_date 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we prove two main results. Firstly, we show that any six-line arrangement, consisting of three pairs of mutually perpendicular lines, does not give rise to a "very generic or sufficiently general" discriminantal arrangement in the sense of C. A. Athanasiadis \citeMR1720104. We give two proofs of the first result. The second result is as follows. The codimension-one boundary faces of (a region) a convex cone of a very generic discriminantal arrangement has not been characterized and is not known even though the intersection lattice of a very generic discriminantal arrangement is known. So secondly, we show that the number of simplex cells of the very generic hyperplane arrangement Hmn=\Hi:\undersetj=1\oversetm∑aijxj=ci,1≤ i≤ n\ may not be not precisely equal to the number of codimension-one boundary hyperplanes of ℝn of the convex cone C containing (c1,c2,…,cn) in the associated very generic discriminantal arrangement. That is, for 1≤ i1