The mod \(p\) rank of incidence matrices for connected uniform hypergraphs
Let \((E,{\mathcal B})\) be an \(r\)-uniform hypergraph. Define the relation \(\approx\) on \(E\) by letting \(x\approx y\) if there is a set \(A\subseteq E- \{x,y\}\) such that \(A\cup\{x\}\in{\mathcal B}\) and \(A\cup\{y\}\in{\mathcal B}\). Now say that \(x\sim y\) if \(x=y\) or \(x=x_ 0\approx x_ 1\approx\cdots\approx x_ s=y\) for some sequence \(x_ 0,\dots,x_ s\in E\). The equivalence relation \(\sim\) is called pivot equivalence. Let \((E,{\mathcal B})\) be a connected \(r\)-uniform hypergraph and let \(E=E_ 1\cup\cdots\cup E_ k\) be the induced partition of \(E\) into pivot equivalence classes. The authors show there exists a multiset \(\{b_ 1,\dots,b_ k\}\) of positive integers such that \(| B\cap E_ i|=b_ i\) for \(i=1,\dots,k\) and all \(B\in{\mathcal B}\). The multiset \(\{b_ 1,\dots,b_ k\}\) is called the multiset of balancing coefficients. The main result then obtained states that if \((E,{\mathcal B})\) is a connected \(r\)-uniform hypergraph over a set \(E\) with \(n\) elements and with balancing coefficients \(b_ 1,\dots,b_ k\) and if \(F\) is a field of characteristic \(p\) (\(p=0\) or \(p\) is a prime) then the rank in \(F\) of the incidence matrix of \({\mathcal B}\) is \(n-k\) if \(p| b_ i\) for all \(i\) and is \(n-k+1\) otherwise. Several significant corollaries follows from this theorem.
scientific article; zbMATH DE number 370656