Polymatroidal ideals and their asymptotic syzygies

Let I be a polymatroidal ideal. In this paper, we study the asymptotic behavior of the homological shift ideals of powers of polymatroidal ideals. We prove that the first homological shift algebra HS₁(R(I)) of I is generated in degree one as a module over the Rees algebra R(I) of I. We conjecture that the ith homological shift algebra HSᵢ(R(I)) of I is generated in degrees ≤ i, and we confirm it in many significant cases. We show that I has the $1$st homological strong persistence property, and we conjecture that the sequence \,HSᵢ(Iᵏ)>0 of associated primes of HSᵢ(Iᵏ) becomes an increasing chain for k≥ i. This conjecture is established when $i=1$ and for many families of polymatroidal ideals. Finally, we explore componentwise polymatroidal ideals, and we prove that HS₁(I) is again componentwise polymatroidal, if I is componentwise polymatroidal.