A solution to the generalized cevian problem using forest polynomials (Q1818216)

The cevian of a (non-degenerate) \(n\)-dimensional simplex is a hyperplane that contains exactly \(n-2\) vertices of the simplex. Assume that for all unordered pairs of vertices \(\{i,j\}\) of a (non-degenerate) \(n\)-dimensional simplex we are given \(x_{ij}\) cevians avoiding vertices \(i\) and \(j\), such that no \(n+1\) cevians intersect in any interior point of the simplex. Then, the paper concludes, the number of domains, into which the cevians partition the interior of the simplex, is the forest polynomial of the complete graph on \(n+1\) vertices, evaluated at the numbers \(x_{ij}\). More precisely, the forest polynomial of the complete graph on \(n+1\) vertices \(1,2,\dots,n,n+1\) is the sum of all forests \(F\) in the complete graph of the forest monomial \(\prod_{\{i,j\}\in F} z_{ij}\) with symmetric indeterminates \(z_{ij}=z_{ji}\), i.e. \(\sum_F\prod_{\{i,j\}\in F}z_{ij}\). At the evaluation \(x_{ij}\) is substituted into \(z_{ij}\).