Strong persistence and associated primes of powers of monomial ideals

Abstract

Let be the polynomial ring in n variables over a field K and I be a monomial ideal of degree . We show that for all and we disprove a motivation question that was asked by of Carlini, Hà, Harbourne and Van Tuyl by providing of a counterexample. Also, by this counterexample, we give a negative answer to the question that the depth function of square-free monomial ideals are non-increasing.