Homological shift ideals of polymatroidal ideals are polymatroidal

Determine whether, for any polymatroid P on [p] with cage m and associated polymatroidal monomial ideal I_P ⊂ R = k[x_1, …, x_p], every homological shift ideal HS_i(I_P) arising from the minimal Z^p-graded free R-resolution of I_P is itself a polymatroidal ideal for all i ≥ 0.

Background

Let P be a polymatroid on [p] with cage m, and I_P ⊂ R the monomial ideal generated by lattice points of its base polytope B(P). The i-th homological shift ideal HS_i(I_P) is defined using shifts appearing in the minimal Z^p-graded free resolution of I_P.

This conjecture, due to Bandari, Bayati, and Herzog, had been verified in several special cases prior to this work: when P is a matroid; when P satisfies the strong exchange property; and when P has rank two. The paper announces that it settles this conjecture in full by proving that all HS_i(I_P) are polymatroidal ideals.