Stanley depth of the integral closure of monomial ideals

摘要

Let I be a monomial ideal in the polynomial ring S=K [x_1,... x_n]. We study the Stanley depth of the integral closure ī of I. We prove that for every integer k ≥ 1, the inequalities (S/I~k) ≤ sdepth (S/ī) and sdepth(I~k) ≤ sdepth(ī) hold. We also prove that for every monomial ideal I? S there exist integers k~1,k~2≥ 1, such that for every s≥ 1, the inequalities sdepth (S/I~(sk1)) ≤ sdepth(S/ī) and sdepth (I~(sk2)) ≤ sdepth (ī) hold. In particular, min_k{sdepth(S/I_k)} ≤ sdepth(S/ī) and min?k {sdepth (I_k)}≤ sdepth(ī). We conjecture that for every integrally closed monomial ideal I, the inequalities sdepth(S/I)≥ n-l (I) and sdepth (I)≥ n-l (I)+1 hold, where l (I) is the analytic spread of I. Assuming the conjecture is true, it follows together with the Burch's inequality that Stanley's conjecture holds for I~k and S/I~k for k ? 0, provided that I is a normal ideal.