Generic-initial-ideal uniqueness by regular-sequence degrees

Prove that if ideals I and J are generated by regular sequences f_1,\dots,f_\ell and g_1,\dots,g_\ell, respectively, with \deg(f_i)=\deg(g_i) for every i, then gin(I)=gin(J) with respect to the reverse lexicographic order.

Background

The paper proves that, for complete intersections of a specified minimal type and sufficiently large smallest generating degree, the generic initial ideal is determined in a prescribed finite range of degrees. The conjecture asks for equality of the entire generic initial ideals whenever the corresponding regular sequences have the same ordered degree sequence, removing the asymptotic and finite-degree restrictions.

The conjecture is presented as a stronger structural statement about how the degrees of a regular sequence determine the reverse-lexicographic generic initial ideal. The paper notes that it would imply the strong Lefschetz property for complete intersections.

References

Let $I$ and $J$ be ideals generated by regular sequences $f_1,\dots, f_\ell$ and $g_1,\dots, g_\ell$ respectively. If for all $i\in {1,\dots, \ell}$ we have $\deg(f_i) = \deg(g_i)$ , then $gin(I) = gin(J)$.

— A Structural Property of Generic Initial Ideals  (2608.17281 - Kuzmanovski, 18 Aug 2026) in Conjecture 6.1, Section 6, “A Few Conjectures”