Search

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let I ⊂ K [ x 1 , ... , x n ] be a monomial ideal and let G(I) denote the (unique) minimal monomial generating set of I. How small can | G (I i) | be in terms of | G (I) | ? Until recently, it was widely expected that | G (I 2) | ≥ | G (I) | would always hold. The first counterexamples emerged in 2018 for n = 2. In this article we show that for any n and d there is an m -primary monomial ideal I ⊂ K [ x 1 , ... , x n ] such that | G (I) | > | G (I i) | for all i ≤ d. [ABSTRACT FROM AUTHOR]