Monomial ideals with tiny squares

Abstract

Let I ⊂ K[x, y] be a monomial ideal. How small can µ(I 2 ) be in terms of µ(I)? It has been expected that the inequality µ(I 2 ) > µ(I) should hold whenever µ(I) ≥ 2. Here we disprove this expectation and provide a somewhat surprising answer to the above question.Theorem 1.1. For every integer m ≥ 5, there exists a monomial ideal I ⊂ K[x, y] such that µ(I) = m and µ(I 2 ) = 9.Moreover, this result is best possible for m ≥ 6.Theorem 1.2. Let I ⊂ K[x, y] be a monomial ideal. If µ(I) ≥ 6 then µ(I 2 ) ≥ 9.Here are some notation to be used throughout. We denote by M the set of monomials in K[x, y], i.e. M = {x i y j | i, j ∈ N}.As usual, we view M as partially ordered by divisibility. For a monomial ideal J ⊂ K[x, y], we denote by G(J) its unique minimal system of monomial generators. It is well known that G(J) is of cardinality µ(J) and consists of all monomials in J which are minimal under divisibility, i.e. G(J) = M ∩ J \ M ∩ J M * where M * = M \ {1}.

Powers of monomial ideals and the Ratliff–Rush operation

“…Note that this bound is no longer valid if I is not equigenerated. Indeed, for each m ≥ 6 there exists a monomial ideal in two variables such that |G(I)| = m and |G(I 2 )| = 9, see [4]. The next proposition is exactly Theorem 2.1 in [9], but we give an alternative proof.…”

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

“…How small can jGðI i Þj be in terms of jGðIÞj? This question has been explored in [1] and [3]. We intuitively expect that the inequality jGðI 2 Þj > jGðIÞj should hold and that jGðI i Þj, i !…”

“…2, grows further whenever jGðIÞj ! 2: This expectation has been disproven in [1]: the authors construct a family of ideals in K½x, y for which jGðIÞj > jGðI 2 Þj:…”

“…In Section 3, we will discuss Theorem 3.1 of [1]. This theorem says that if a monomial ideal I ¼ hu 1 , :::, u m i & K½x, y satisfies certain conditions, then jGðI 2 Þj ¼ 9: We will relax the conditions of this theorem and give a more intuitive proof.…”

Freiman ideals

“…The assumption that I be equigenerated is essential. Indeed, in [4] it has been shown that for every integer m ≥ 6, there exists a monomial ideal I ⊂ K[x, y] such that µ(I) = m and µ(I 2 ) = 9.…”