On matroids which have precisely one basis in common
type: Item
instance of: scholarly article
title: On matroids which have precisely one basis in common
zbMATH Open document ID: 0663.05019
DOI: 10.1016/S0195-6698(88)80039-8
author: Andreas W. M. Dress
published in: European Journal of Combinatorics
publication date: 1988-01-01T00:00:00Z
review text: A theorem concerning matroids is proved which - specialized to representable matroids - implies that given a non-singular quadratic \(n\times n\)-matrix \(A=(a_{ij})\) and some \(k\in \{1,...,n-1\}\) such that for any subset \(S\subseteq \{1,...,n\}\) of cardinality k different from \(\{\) 1,...,k\(\}\), the product \(\det ((a_{ij})_{i=1,...,k;j\in S})\cdot \det ((a_{ij})_{i=k+1,...,n;j\not\in S})\) in the Laplace expansion of A with respect to the first k and the last n-k rows vanishes, there exists some \(i\in \{k+1,...,n\}\) with \(a_{1i}=a_{2i}=...=a_{ki}=0\) or some \(i\in \{1,...,k\}\) with \(a_{k+1,i}=a_{k+2,i}=...=a_{ni}=0.\)
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 4083644
zbMATH Keywords: matroids
zbMATH Keywords: representable matroids
zbMATH Keywords: Laplace expansion
MaRDI profile type: Publication
cites work: Q4111952
full work available at URL: s0195-6698(88)80039-8
OpenAlex ID: W1976284042
description: scientific article; zbMATH DE number 4083644