Monomial ideals with arbitrarily high tiny powers in any number of variables

Abstract

Powers of (monomial) ideals is a subject that still calls attraction in various ways. Let I & K½x 1 , :::, x n be a monomial ideal and let G(I) denote the (unique) minimal monomial generating set of I. How small can jGðI i Þj be in terms of jGðIÞj? Until recently, it was widely expected that jGðI 2 Þj ! jGðIÞj would always hold. The first counterexamples emerged in 2018 for n ¼ 2. In this article we show that for any n and d there is an m-primary monomial ideal I & K½x 1 , :::, x n such that jGðIÞj > jGðI i Þj for all i d: ARTICLE HISTORY

Rooted order on minimal generators of powers of some cover ideals

“…In [4], the authors addressed the question of how small µ(I 2 ) can be in terms of µ(I) when I is a monomial ideal in polynomial ring with n = 2 variables. Behaviour of µ(I s ) was considered in some other articles, see for example [1,10,16,17]. Recently, Drabkin and Guerrieri [3] studied Freiman cover ideals.…”