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.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)|? We expect that the inequality |G(I 2 )| > |G(I)| should hold and that |G(I i )|, i ≥ 2, grows further whenever |G(I)| ≥ 2. In this paper we will disprove this expectation and 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.

Certain monomial ideals whose numbers of generators of powers descend

“…Let us explain about this result. The inequalities µ(I) > µ(I 2 ) > • • • > µ(I n ) are more unexpected than the behavior obtained in [5], and the equality µ(I n ) = (n+1) 2 is related to the lower bound of [4, Theorem 1.2] regarding (n + 1) 2 as 9 by n = 2. The latter assertion claims µ(I k ) agrees with the polynomial for k ≥ n. Thus our construction provides one of the most unexpected numbers of generators of monomial ideals before being a polynomial function.…”

“…In particular, the function diverges infinitely. However, before being a polynomial, the numbers of generators of powers sometimes behave in an unexpected way even in polynomial rings (see for example [2] and [5]).…”

“…Let K[x, y] be the polynomial ring over a field K and I a monomial ideal of K[x, y]. Then Eliahou, Herzog, and Saem [4, Theorem 1.2] showed a sharp lower bound µ(I 2 ) ≥ 9 when µ(I) ≥ 6, and Gasanova [5] construct monomial ideals such that µ(I) > µ(I n ) for any positive integer n. With this background, our result is stated as follows.…”

“…The latter assertion claims µ(I k ) agrees with the polynomial for k ≥ n. Thus our construction provides one of the most unexpected numbers of generators of monomial ideals before being a polynomial function. We also note that our ideal to describe Theorem 1.1 are constructed by n equigenerated ideals in contrast that [4,5] and [2] basically explore ideals constructed by two equigenerated ideals.…”

On the initial behaviour of the number of generators of powers of monomial ideals

“…Quite to the contrary, if the monomial ideal I is not generated in a single degree, then it may very well happen that I 2 has less generators than I. For monomial ideals I in 2 variable, a sharp lower bound for the number of generators µ(I 2 ) of I 2 is given in [3], and Gasanova [4] gave examples of monomial ideals I with the property that for any given number k one has µ(I k ) < µ(I).…”