Stanley Depth of the Integral Closure of Monomial Ideals

Let I be a monomial ideal in the polynomial ring S = K[x1, . . . , xn]. We study the Stanley depth of the integral closure I of I. We prove that for every integer k ≥ 1, the inequalities sdepth(S/Ik) ≤ sdepth(S/I) and sdepth(Ik) ≤ sdepth(I) hold. We also prove that for every monomial ideal I ⊂ S there exist integers k1, k2 ≥ 1, such that for every s ≥ 1, the inequalities sdepth(S/I1) ≤ sdepth(S/I) and sdepth(I2) ≤ sdepth(I) hold. In particular, mink{sdepth(S/I)} ≤ sdepth(S/I) and mink{sdepth(I)} ≤ sdepth(I). We conjecture that for every integrally closed monomial ideal I, the inequalities sdepth(S/I) ≥ n−`(I) and sdepth(I) ≥ n−`(I)+1 hold, where `(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 and S/I for k 0, provided that I is a normal ideal.