Regularity of symbolic powers of polymatroidal ideals

Determine whether every polymatroidal ideal I in a polynomial ring satisfies reg I^(k) = reg I^k for all integers k ≥ 1.

Background

The paper studies the relationship between symbolic and ordinary powers of polymatroidal ideals. Ordinary powers of polymatroidal ideals have linear resolutions, whereas the regularity of symbolic powers is generally more difficult to control. Conjecture A proposes that the regularity of every symbolic power agrees with that of the corresponding ordinary power, which would imply eventual linearity of the symbolic-power regularity function.

The paper proves this equality for several families, including squarefree Veronese ideals, matching-matroidal ideals of Veronese type, transversal polymatroidal ideals, principal Borel ideals, polymatroidal ideals generated in degree two, polymatroidal ideals in at most three variables, and matroidal ideals in at most four variables. It therefore remains unresolved in the full class of polymatroidal ideals.