Total anisotropy on the moment curve in characteristic p

Establish that, for every field k of characteristic p and every (d−1)-dimensional simplicial cycle μ over k, there exists a transcendental field extension k′ of k and an Artinian reduction of the graded commutative face ring k[|μ|] that is anisotropic: every nonzero element u of degree k≤d/2 in the associated Gorensteinification B^k(μ) satisfies u²≠0.

Background

The paper proves p^m-anisotropy for simplicial cycles satisfying a link-homology condition, using Artinian reductions whose linear systems of parameters correspond to points on the moment curve. This resolves earlier questions about extending characteristic-two anisotropy to p-anisotropy and about restricting the choice of parameters.

The unresolved conjecture asks whether ordinary total anisotropy, namely nonvanishing of squares of all nonzero elements through the middle degree, holds over a suitable transcendental extension in arbitrary characteristic p for arbitrary simplicial cycles, without the additional condition used in the main theorem. The authors state that their derivative-based approach did not settle this question and might have failed because the conjecture is false or because a different method is needed.