Heuristic Algorithms for Recognition of Some Cubic Hypersurfaces

In this paper, we propose some heuristic probabilistic polynomial time algorithms with one-sided error for recognition of cubic hypersurfaces the singular loci of which do not contain any linear subspace of sufficiently large dimension. These algorithms are easy to implement in computer algebra systems. The algorithms are based on checking the condition that the Hessian determinant of a cubic form does not vanish identically or does not determine any cone in the projective space. In turn, the properties of the Hessian can be verified with one-sided-error probabilistic algorithms based on the Schwartz–Zippel lemma.