The generic anisotropy of simplicial 1-spheres (Q6161734)

Let \(D\) be a simplicial sphere of dimension \(d-1\) on a set of \(m\) vertices and let \(k_1\) be a field. Take \(d\times m\) (coefficient) variables, \(c_{ij}\), and form the field \[ k=k_1(c_{ij}: 1\leq i\leq d, 1\leq j\leq m). \] Let \(I_D\subseteq R=k[x_1,\dots,x_m]\) denote the Stanley-Reisner ideal of \(D\) and, for every \(i=1,\dots,d\), denote \[ f_i=\sum_{j=1}^m a_{ij}x_j. \] The authors define the generic Artinian reduction of the Stanley-Reisner ring of \(D\) to be the quotient \(A=R/(I_D,f_1,\dots,f_d)\). They call \(D\) generically anisotropic over \(k_1\) if, for all \(j\) such that \(1\leq 2j\leq d\) and all \(0\not = u \in A_j\), \(u^2\not =0\). The \textit{Anisotropy conjecture} states that any simplicial sphere has this property. The main result in this paper (Theorem 1.3) is the proof of the Anisotropy conjecture in the one-dimensional case.