Optimality of the deterministic GIN-position algorithm

Determine whether Algorithm 1, which uses generic upper-unitriangular coordinate transformations and rank or maximal-minor conditions to compute a generic initial ideal, is the most efficient algorithm currently available that guarantees the correct generic initial ideal.

Background

The paper develops a deterministic linear-algebra algorithm for placing a homogeneous polynomial ideal in generic initial ideal (GIN) position. The procedure applies a generic upper-unitriangular coordinate transformation, constructs coefficient matrices for the homogeneous components up to the Castelnuovo–Mumford regularity, and determines nonvanishing conditions on maximal minors that guarantee the transformed ideal has the generic initial ideal.

Although the authors provide a complexity analysis and describe the algorithm as computationally expensive, they do not establish that it is optimal or even most efficient among all currently available guaranteed methods. The unresolved issue is therefore a comparative algorithmic one: whether the proposed deterministic procedure achieves the best known efficiency while retaining a guarantee of correctness, rather than relying on random coordinate changes.