Linear resolutions and polymatroidal ideals

Let R = K [ x 1 , … , x n ] be the polynomial ring in n variables over a field K and I be a monomial ideal generated in degree d . Bandari and Herzog ( Eur. J. Combin. 34 (2013) 752–763) conjectured that a monomial ideal I is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper, we give an affirmative answer to the conjecture in the following cases: (i) height ( I ) = n - 1 ; (ii) I contains at least n - 3 pure powers of the variables x 1 d , … , x n - 3 d ; (iii) I is a monomial ideal in at most four variables.