Growth of primary decompositions of Frobenius powers of ideals

It was previously known, by work of Smith–Swanson and of Sharp–Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth property holds. For non-negatively graded rings we are able to show that there is a “polynomial growth”. We present explicit primary decompositions of Frobenius powers of an ideal, which were known to have infinitely many associated primes, having this linear growth property. We also discuss some other interesting examples.