Nonemptiness of the reverse-lexicographic genericity locus U_RL

Determine whether the Zariski open subset U_RL, constructed as the intersection of the loci ensuring the prescribed Hilbert series and the rank conditions on the leftmost columns of the Macaulay matrices for I_{n−2}(M), is nonempty; equivalently, show that for each degree d with n − 1 ≤ d ≤ 2n − 3, the determinant polynomials g_d(a) of the specified square submatrices of the Macaulay matrix M_d(𝔽) are nonzero in k[a].

Background

To support the conjectured generic reverse lexicographic property, the authors construct a specific Zariski open subset U_RL by intersecting conditions: (i) the Hilbert series matches the form derived from the Gulliksen–Negård complex, and (ii) each reduced Macaulay matrix in degrees d = n − 1, …, 2n − 3 has its first h_d columns forming a full-rank block (equivalently, its echelon form begins with an identity block). These rank conditions correspond to nonvanishing determinants g_d(a) of certain square submatrices.

The authors explicitly note that it is not clear whether U_RL is nonempty, which reduces to checking that all g_d(a) are nonzero polynomials in the specialization parameters a. Resolving this would validate the reverse lexicographic structure generically and underpin the complexity estimates that rely on this structure.