Homological shift ideals of polymatroidal ideals

Determine whether for every polymatroid P on the ground set [p] with cage m ∈ N^p, the homological shift ideals HS_i(I_P) arising from the minimal Z^p-graded free resolution of the polymatroidal ideal I_P ⊂ k[x_1,…,x_p] (generated by monomials corresponding to lattice points of the base polytope B(P)) are themselves polymatroidal ideals for all integers i ≥ 0.

Background

The paper studies polymatroidal ideals associated to a polymatroid P and introduces the cave polynomial to connect K-theoretical and combinatorial invariants. A central question posed in prior work by Bandari, Bayati, and Herzog asks whether all homological shift ideals of a polymatroidal ideal remain polymatroidal.

Partial confirmations were known: for matroids (Bayati–Herzog), for polymatroids with the strong exchange property (Herzog–Mohammadi), and for rank-two polymatroids (Ficarra), with related developments in subsequent works. The present paper states the conjecture explicitly and later claims to settle it.