Resumen de Stanley depth of the integral closure of monomial ideals

S. A. Seyed Fakhari

• Let ? be a monomial ideal in the polynomial ring ?=?[?1,…,??] . We study the Stanley depth of the integral closure ?⎯⎯⎯ of ? . We prove that for every integer ?≥1 , the inequalities sdepth(?/??⎯⎯⎯⎯⎯⎯)≤ sdepth(?/?⎯⎯⎯) and sdepth(??⎯⎯⎯⎯⎯⎯)≤ sdepth(?⎯⎯⎯) hold. We also prove that for every monomial ideal ?⊂? there exist integers ?1,?2≥1 , such that for every ?≥1 , the inequalities sdepth(?/???1)≤ sdepth(?/?⎯⎯⎯) and sdepth(???2)≤ sdepth(?⎯⎯⎯) hold. In particular, min?{ sdepth(?/??)}≤ sdepth(?/?⎯⎯⎯) and min?{ sdepth(??)}≤ sdepth(?⎯⎯⎯) . We conjecture that for every integrally closed monomial ideal ? , the inequalities sdepth(?/?)≥?−ℓ(?) and sdepth(?)≥?−ℓ(?)+1 hold, where ℓ(?) is the analytic spread of ? . Assuming the conjecture is true, it follows together with the Burch’s inequality that Stanley’s conjecture holds for ?? and ?/?? for ?≫0 , provided that ? is a normal ideal.