Critical binomial ideals of Norhtcott type

TL;DR

This paper investigates a family of binomial ideals defining monomial curves in affine space, proving they are set-theoretic complete intersections and computing key invariants for irreducible cases.

Contribution

It introduces a new class of binomial ideals related to monomial curves, proves their set-theoretic complete intersection property, and provides a method to generate high-height semigroup ideals.

Findings

Monomial curves in the family are set-theoretic complete intersections.

Computed invariants like genus, type, and Frobenius number for irreducible cases.

Method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

Abstract

In this paper, we study a family of binomial ideals defining monomial curves in the $n-$dimensional affine space determined by $n$ hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}} \in k[x_1, \ldots, x_n]$ with $u_{ii} = 0$, $i\in \{ 1, \ldots, n\}$. We prove that, the monomial curves in that family are set-theoretic complete intersection. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Fr\"obenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

Equations

$$\pi :\mathbb{k}[\mathbf{x}]\to \mathbb{k}[\mathcal{A}]:={⨁}_{\mathbf{a}\in \mathbb{N}\mathcal{A}}\mathbb{k}\{{\mathbf{t}}^{\mathbf{a}}\};x_i\mapsto {\mathbf{t}}^{{\mathbf{a}}_i},$$

$$A=\left(\begin{array}[]{ccccc}3&5&7&0&0\\ 0&0&0&2&3\end{array}\right)$$

$$B=\left(\begin{array}[]{cccc}c_{1}&0&-u_{13}&-u_{14}\\ -u_{21}&c_{2}&0&-u_{24}\\ 0&-u_{32}&c_{3}&-u_{34}\\ -u_{41}&-u_{42}&-u_{43}&c_{4}\end{array}\right).$$

$$\begin{array}[]{ccc}f_{1}&=&a_{11}m_{1}+a_{12}m_{2}+\ldots+a_{1d}m_{d},\\ f_{2}&=&a_{21}m_{1}+a_{22}m_{2}+\ldots+a_{2d}m_{d},\\ &\vdots&\\ f_{d}&=&a_{d1}m_{1}+a_{d2}m_{2}+\ldots+a_{dd}m_{d},\end{array}$$

$$\Phi=\left(\begin{array}[]{ccc}x_{1}^{u_{41}}&0&-x_{4}^{u_{14}}\\ -x_{4}^{u_{24}}&x_{2}^{u_{42}}&0\\ 0&-x_{4}^{u_{34}}&x_{3}^{u_{43}}\end{array}\right)$$

$$\Phi=\left(\begin{array}[]{cccc}x_{1}^{u_{n1}}&0&\ldots&-x_{n}^{u_{1n}}\\ -x_{n}^{u_{2n}}&x_{2}^{u_{n2}}&\ldots&0\\ 0&-x_{n}^{u_{3n}}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&x_{n-1}^{u_{nn-1}}\end{array}\right)$$

$$M:=\left(\begin{array}{cccccc}{u}_{n1}+u_{21}& 0& \dots & 0& -u_{1n-1}& -u_{1n}\\ -u_{21}& u_{n2}+u_{32}& \dots & 0& 0& -u_{2n}\\ 0& -u_{32}& \dots & 0& 0& -u_{3n}\\ ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& \dots & -u_{n-1n-2}& u_{nn-1}+u_{1n-1}& -u_{n-1n}\\ -u_{n1}& -u_{n2}& \dots & -u_{nn-2}& -u_{nn-1}& {\sum }_{i=1}^{n-1}{u}_{in}\end{array}\right)$$

$$\left({\prod }_{i=1}^{n-2}(u_{ni}+u_{i+1i})\right)\left(u_{nn-1}+u_{1n-1}\right)-\left({\prod }_{i=1}^{n-2}{u}_{i+1i}\right)u_{1n-1}>0,$$

$${\alpha }_1{u}_{n1}+({\alpha }_1-{\alpha }_2)u_{21}$$

$$⋮$$

$${\alpha }_{n-1}{u}_{nn-1}+({\alpha }_{n-1}-{\alpha }_1)u_{1n-1}$$

$${\sum }_{i=1}^{n-1}(-{\alpha }_i)u_{in}$$

$$S(f_i,D)=$$

$$+x_{{\sigma }^{n-2}(i)}^{u_{n{\sigma }^{n-2}(i)}}\cdots x_{{\sigma }^3(i)}^{u_{n{\sigma }^3(i)}}{x}_n^{u_{in}+u_{\sigma (i)n}}{f}_{{\sigma }^2(i)}+$$

$$+\cdots +$$

$$+x_n^{u_{in}+\cdots +u_{{\sigma }^{n-3}(i)n}}{f}_{{\sigma }^{n-2}(i)}$$

$$(J:x_i^{s_i-u_{ni}})$$

$$=\left((J:x_i^{s_i})\cap (J+⟨x_i^{s_i}⟩):x_i^{s_i-u_{ni}}{x}_n^{{\sum }_{i=1}^{n-1}{u}_{in}}\right)$$

$$=\left((J:x_i^{s_i}):x_i^{s_i-u_{ni}}{x}_n^{{\sum }_{i=1}^{n-1}{u}_{in}}\right)\cap \left((J+⟨x_i^{s_i}⟩):x_i^{s_i-u_{ni}}{x}_n^{{\sum }_{i=1}^{n-1}{u}_{in}}\right)$$

$$=(J:x_i^{2s_i-u_{ni}}),$$

$$J$$

$$=\left((J:x_i^{s_i}):x_n^{{\sum }_{i=1}^{n-1}{u}_{in}}\right)\cap \left((J+⟨x_i^{s_i}⟩):x_n^{{\sum }_{i=1}^{n-1}{u}_{in}}\right)=(J:x_i^{s_i})$$

$$I_{\mathcal{A}}=I_{+}(\rho):=\big{\langle}\{\mathbf{x}^{\mathbf{v}_{+}}+\rho(\mathbf{v})\mathbf{x}^{\mathbf{v}_{-}}\ \mid\ \mathbf{v}\in\mathcal{M}\}\big{\rangle},$$

$$\mathrm{Sat}(\mathcal{L}):=\{\mathbf{v}\in {\mathbb{Z}}^n\mid d\mathbf{v}\in \mathcal{L}\text{mbox}forsomed\in \mathcal{L}\setminus \{0\}\}.$$

$$\sqrt{I_{+}(\rho )}=I_{+}({\rho }^{\prime }),$$

$$\mathrm{Ass}(S/I_{+}(\rho ))=\{I_{+}({\rho }_j^{\prime })\mid j=1,\dots ,g\}$$

$$I_{+}(\rho )=\bigcap _{j=1}^g{I}_{+}({\rho }_j)$$

$$\Phi=\left(\begin{array}[]{cccc}x_{1}^{2}&0&-x_{4}\\ -x_{4}^{2}&x_{2}^{2}&0\\ 0&-x_{4}&x_{3}^{4}\end{array}\right)$$

$$M=\left(\begin{array}[]{rrrr}3&0&-1&-1\\ -1&4&0&-2\\ 0&-2&5&-1\\ -2&-2&-4&4\end{array}\right).$$

$$I_{\mathcal{A}}=⟨x_1^3-x_3{x}_4,x_2^4-x_1{x}_4^2,x_3^5-x_2^2{x}_4,x_1^2{x}_2^2{x}_3^4-x_4^4⟩\subset \mathbb{k}[x_1,\dots ,x_n],$$

$$\{x_1^3-x_3{x}_4,x_2^4-x_1{x}_4^2,x_3^5-x_2^2{x}_4,x_1{x}_2{x}_3^2\pm x_4^2,x_2{x}_3^3\pm x_1^2{x}_4\}.$$

$$\gcd \left(\prod _{i=1}^{n-1}(u_{ni}+1)-1,u_{1n}+\sum _{i=1}^{n-2}\left(\prod _{j=1}^i(u_{nj}+1)\right)u_{i+1n}\right).$$

$$M:=\left(\begin{array}{ccccccc}{u}_{n1}+1& 0& 0& \dots & 0& -1& -u_{1n}\\ -1& u_{n2}+1& 0& \dots & 0& 0& -u_{2n}\\ 0& -1& u_{n3}+1& \dots & 0& 0& -u_{3n}\\ ⋮& ⋮& ⋮& & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & -1& u_{nn-1}+1& -u_{n-1n}\\ -u_{n1}& -u_{n2}& -u_{n3}& \dots & -u_{nn-2}& -u_{nn-1}& {\sum }_{i=1}^{n-1}{u}_{in}\end{array}\right).$$

$$(u_{nk}+1)a_k-a_{{\sigma }^{-1}(k)}-u_{kn}{a}_n=0,$$

$$\mathcal{A}=\left\{2^{n-1}+\sum _{k=0}^{n-3}{2}^k,2^{n-1}+\sum _{k=0}^{n-4}{2}^k,\dots ,2^{n-1}+1,2^{n-1},2^{n-1}-1\right\}$$

Full text

The first and second authors are supported by the project MTM2014-55367-P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andalucía Grant Number FQM-343

The third author is partially supported by the project MTM2015-65764-C3-1, National Plan I+D+I and by Junta de Extremadura (FEDER funds) - FQM-024.

Introduction

We study a family of unit free monoids contained in $T\otimes\mathbb{Z}$, with $T$ a finite Abelian group. The monoids of this family have an associated ideal generated by critical binomials, and they extend several classes of numerical semigroups studied by Bresinsky, Alcántar and Villareal, Herzog, O’Carrol and Planas, among others [1, 3, 12, 18]. These monoids will be called semigroups of Northcott type, since their defining ideals are inspired in [17].

Ideals of Northcott type are defined as the ideal generated by the coordinates of the product of a matrix $\Phi$ and a vector $\mathbf{m}$. The matrix $\Phi$ has all its entries equal to zero except in the diagonal and below the diagonal (and the upper right corner); nonzero entries are monomials in a single variable. The vector $\mathbf{m}$ is also a vector of monomials in each of the variables. We show that every ideal of Northcott type is realizable as the defining ideal of a monoid of rank one. By gluing these monoids with copies of $\mathbb{N}$, we cover and extend the families studied in [1].

We are able to prove that ideals of Northcott type have a unique (up to signs) minimal system of binomial generators. This in particular means that monoids of Northcott type are uniquely presented. We also depict how the primary decomposition of these ideals looks like.

As monoids of Northcott type are uniquely presented, we are able to derive some nonunique factorization properties. We show what are the maximum and minimum of the Delta sets of factorizations of the monoid, as well as the catenary degree.

As a particular case, we study the case when our monoids are numerical semigroups. We show how to find families of Northcot type ideals yielding numerical semigroups, and compute some relevant invariants in terms of the exponents of the monomials appearing in $\Phi$ and $\mathbf{m}$. We start by computing the Apéry sets, and from these we easily derive formulas for the genus, Frobenius number and pseudoFrobenius numbers (and thus the type) of the semigroup.

We show that if we have a monoid with defining ideal a critical ideal, and we glue this monoid with a copy of $\mathbb{N}$, then the resulting monoid is again critical. As we mentioned above, this allows us to cover several families studied in the literature that appear as a gluing of a Northcott type monoid and a copy of $\mathbb{N}$. One can repeat this process and obtain a family of critical monoids. Not every critical monoid can be obtained in this way.

In the final section we show that every ideal of Northcott type is a set-theoretic complete intersection. As gluings by copies of $\mathbb{N}$ preserve set-theoretic complete intersections, the monoids of the family considered in this manuscript are set-theoretic complete intersections.

1. Preliminaries

Let $G$ be a finitely generated commutative group and set $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$. Let $\mathbb{N}\mathcal{A}:=\mathbb{N}\mathbf{a}_{1}+\dots+\mathbb{N}\mathbf{a}_{n}$ be the commutative monoid generated by $\mathcal{A}$, where $\mathbb{N}$ stands for the set of non-negative integers. Throughout this paper, we assume that $\mathcal{A}$ is chosen to have the following property $\mathbb{N}\mathcal{A}\cap(-\mathbb{N}\mathcal{A})=\{0\}$, that is to say, $\mathbb{N}\mathcal{A}$ is supposed to be unit free.

Let $\Bbbk[\mathbf{x}]:=\Bbbk[x_{1},\ldots,x_{n}]$ be the polynomial ring in $n$ variables over an arbitrary field $\Bbbk$. As usual, we will denote by $\mathbf{x}^{\mathbf{u}}$ the monomial $x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$ of $\Bbbk[\mathbf{x}],$ where $\mathbf{u}=(u_{1},\ldots,u_{n})\in\mathbb{N}^{n}$. The ring $\Bbbk[\mathbf{x}]$ can be graded by $\mathbb{N}\mathcal{A}$ by assigning degree $\mathbf{a}_{i}$ to the variable $x_{i}$, $i\in\{1,\ldots,n\}$. So, by abusing the notation slightly, we will say that $\Bbbk[\mathbf{x}]$ is $\mathcal{A}-$graded.

Consider the ring homomorphism

[TABLE]

where the product of $\Bbbk[\mathcal{A}]$ is given by $\mathbf{t}^{\mathbf{a}}\cdot\mathbf{t}^{\mathbf{a}^{\prime}}=\mathbf{t}^{\mathbf{a}+\mathbf{a}^{\prime}}$. By [16, Lemma 7.3], the kernel of $\pi$ is the ideal generated by all the binomials $\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}$ such that $\pi(\mathbf{x}^{\mathbf{u}})=\pi(\mathbf{x}^{\mathbf{v}})$. This binomial ideal is usually called the semigroup ideal of $\mathcal{A}$ and it is denoted by $I_{\mathcal{A}}$. Clearly, $\Bbbk[\mathcal{A}]$ is $\mathcal{A}-$graded and $I_{\mathcal{A}}$ is $\mathcal{A}-$homogeneous.

It is convenient to observe that $I_{\mathcal{A}}$ is not necessarily a prime ideal. In fact, $I_{\mathcal{A}}$ is prime if and only if $\mathbb{Z}\mathcal{A}:=\mathbb{Z}\mathbf{a}_{1}+\dots+\mathbb{Z}\mathbf{a}_{n}$ is torsion free (this will happen, in particular, when $G$ has not torsion). In this case, $I_{\mathcal{A}}$ is a so-called toric ideal and the monoid $\mathbb{N}\mathcal{A}$ is said to be an affine semigroup.

Recall that the dimension of the zero set of $I_{\mathcal{A}}$ is equal to the rank of $\mathbb{Z}\mathcal{A}$ (see, e.g., [16, Proposition 7.5]).

Definition 1.1**.**

With the above notation, a binomial $x_{i}^{c_{i}}-\prod_{j\neq i}x_{j}^{u_{ij}}\in I_{\mathcal{A}}$ is called critical with respect to $x_{i}$ if $c_{i}$ is the least positive integer such that $c_{i}\mathbf{a}_{i}\in\sum_{j\neq i}\mathbb{N}\mathbf{a}_{j}$. A subideal of $I_{\mathcal{A}}$ generated by $n$ critical binomials, one for each variable, is said to be a critical binomial ideal associated to $\mathcal{A}$.

Our first result gives a sufficient condition for the existence of critical binomial ideals.

Proposition 1.2**.**

If $\mathbb{Z}\mathcal{A}$ has rank $1$, then $I_{\mathcal{A}}$ contains a critical binomial ideal associated to $\mathcal{A}$.

Proof.

By the fundamental theorem of finitely generated commutative groups $\mathbb{Z}\mathcal{A}\cong\mathbb{Z}\oplus T$ for some finite (abelian) group $T$. Let $t\geq 0$ be the order of $T$. Thus, we may suppose that $\mathbf{a}_{i}=(a_{i},\mathbf{b}_{i})$ such that $a_{i}\in\mathbb{Z}$ and $\mathbf{b}_{i}\in T$, $i\in\{1,\ldots,n\}$. Clearly, $a_{1}a_{i}\in\sum_{j\neq i}\mathbb{N}a_{j},$ $i\in\{2,\ldots,n\}$ and $a_{n}a_{1}\in\sum_{j\neq 1}\mathbb{N}a_{j}$, so it follows that $(ta_{1})\mathbf{a}_{i}\in\sum_{j\neq i}\mathbb{N}\mathbf{a}_{j}$, $i\in\{2,\ldots,n\}$ and $(ta_{n})\mathbf{a}_{1}\in\sum_{j\neq 1}\mathbb{N}\mathbf{a}_{j}$. Therefore, for each $i$, there exists a minimal $c_{i}>0$ such that $c_{i}\mathbf{a}_{i}\in\sum_{j\neq i}\mathbb{N}\mathbf{a}_{j}$. Thus, $I_{\mathcal{A}}$ contains a critical binomial ideal. ∎

The converse of the above proposition need not be true in general.

Example 1.3**.**

Let $\mathbf{a}_{i}$ be the $i$th column of the following matrix

[TABLE]

and set $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{5}\}\subseteq\mathbb{Z}^{2}$. By direct computation, one can check that $I_{\mathcal{A}}$ is generated by critical binomials.

Definition 1.4**.**

With the above notation, we will say that $\mathbb{N}\mathcal{A}$ is a critical monoid if $I_{\mathcal{A}}$ is a critical binomial ideal associated to $\mathcal{A}$. In this case, for short, we will say that the binomial ideal $I_{\mathcal{A}}$ is critical.

In the following, we will assume that $\mathbb{Z}\mathcal{A}$ has rank one; in particular $I_{\mathcal{A}}$ will contain a critical binomial ideal associated to $\mathcal{A}$ by Proposition 1.2. One of the aims of this work is to give conditions for $\mathbb{N}\mathcal{A}$ to be a critical monoid; equivalently, for $I_{\mathcal{A}}$ to be critical. For this purpose, we summarize first the known results:

If $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subseteq\mathbb{N}$ and $\mathrm{gcd}(\mathbf{a}_{1},\ldots,\mathbf{a}_{n})=1$, then $\mathbb{N}\mathcal{A}$ is a numerical semigroup and $I_{\mathcal{A}}$ is the defining ideal of an irreducible monomial curve. In this case, it is known that every minimal system of binomial generators of $I_{\mathcal{A}}$ contains, up to sign, a critical binomial with respect to $x_{i},$ for each $i$ (see, for example, [13, Proposition 2.3]).

Under the hypothesis of $\mathbb{N}\mathcal{A}$ being a numerical semigroup, Herzog proved that every (toric) ideal $I_{\mathcal{A}}$ in $\Bbbk[x_{1},x_{2},x_{3}]$ is critical (see [12, Section 3], or [20, Theorem 2.2] for a concise formulation of Herzog’s results). For $n=4,$ Alcántar and Villarreal proved in [1] that if $I_{\mathcal{A}}$ is critical of height $3,$ then (after permuting the variables appropriately) it has a set of binomial generators of one of the following three types.

• (N):

$f_{1}=x_{1}^{c_{1}}-x_{3}^{u_{13}}x_{4}^{u_{14}},f_{2}=x_{2}^{c_{2}}-x_{1}^{u_{21}}x_{4}^{x_{24}},f_{3}=x_{3}^{c_{3}}-x_{2}^{u_{32}}x_{4}^{u_{34}}$ and $D=x_{4}^{c_{4}}-x_{1}^{u_{41}}x_{2}^{u_{42}}x_{3}^{u_{43}},$ with $u_{ij}>0,$ for all $i,j$.

• (gN1):

$f_{1}=x_{1}^{c_{1}}-x_{3}^{c_{3}},f_{2}=x_{2}^{c_{2}}-x_{1}^{u_{21}}x_{3}^{u_{23}}$ and $g=x_{4}^{c_{4}}-x_{1}^{u_{41}}x_{2}^{u_{42}}x_{3}^{u_{43}}$.

• (gN2):

$f_{1}=x_{1}^{c_{1}}-x_{2}^{u_{12}}x_{3}^{u_{13}},f_{2}=x_{2}^{c_{2}}-x_{1}^{u_{21}}x_{3}^{u_{23}},D=x_{3}^{c_{3}}-x_{1}^{u_{31}}x_{2}^{u_{32}}$ and $g=x_{4}^{c_{4}}-x_{1}^{u_{41}}x_{2}^{u_{42}}x_{3}^{u_{43}},$ with $u_{ij}>0$ for all $i\neq 4$ and $u_{4j}\geq 0,$ not all zero, for all $j$.

Moreover, under suitable arithmetically conditions on the exponents, the converse turns out to be true (see Propositions 1.5 and 5.1).

The following result due to Bresinsky ([3, Theorem 5]) gives a necessary and sufficient condition for the critical ideal generated by (N) to be equal to $I_{\mathcal{A}}$.

Proposition 1.5**.**

With the same notation as in (N), let $B$ be the matrix whose rows are the exponent vectors of $f_{1},f_{2},f_{3}$ and $D$, that is,

[TABLE]

Then there exists $\mathcal{A}\subset\mathbb{N}$ such that $\mathbb{N}\mathcal{A}$ is a numerical semigroup and $I_{\mathcal{A}}=\langle f_{1},f_{2},f_{3},D\rangle$ if and only if

• (a)

the sum of the columns of $B$ are zero;

• (b)

*the *$(4,j)-$minors of $B$, $j\in\{1,\ldots,4\}$ are relatively prime.

Proof.

See [1, Proposition 3.2] for a proof in our notation. ∎

Notice that conditions (a) and (b) in the above proposition implies that $B$ has rank $3$. So, there exist $a_{1},\ldots,a_{4}\in\mathbb{N}$ relatively prime such that $(a_{1},\ldots,a_{4})B=0$. In particular, we have that the $(4,j)-$minor of $B$ is equal to $\alpha a_{j}$ for some nonzero integer $\alpha$ which is necessarily equal to one by condition (b).

2. Critical ideals of Northcott type

In this section, we will generalize the family (N) for arbitrary $n$ by producing a family of critical binomial ideal of Northcott type of the form $I_{\mathcal{A}}$. This will allow us to determine whole families of numerical semigroups whose semigroup ideal associated is critical.

It is fair to point out that the case $n=3$ has been analyzed exhaustively by O’Carroll and Planas-Vilanova in [18] for a Noetherian ring $R$ and $\mathbf{m}=(x_{1},\ldots,x_{n})$ being a sequence of elements of $R$ generating an ideal of height $n$.

Now, for convenience, we reproduce verbatim (but adapted to our notation) the definition of Northcott ideal given in [18]: “In [17], Northcott considered the following situation: let $\mathbf{m}=(m_{1},\ldots,m_{d})$ and $\mathbf{f}=(f_{1},\ldots,f_{d})$ be two sets of $d$ elements of a Noetherian ring $R,$ connected by the relations

[TABLE]

with $a_{ij}\in R$. Let $D$ stand for the determinant of the $d\times d$ matrix $\Phi=(a_{ij})$. Northcott proved that if $\langle f_{1},\ldots,f_{d}\rangle$ has grade $d$ and $\langle f_{1},\ldots,f_{d},D\rangle$ is proper, then the projective dimension of $R/\langle f_{1},\ldots,f_{d},D\rangle$ is $d,$ and $\langle f_{1},\ldots,f_{d},D\rangle$ and all its associated prime ideals have grade $d$ [17, Theorem 2]. Subsequently, Vasconcelos called such an ideal $\langle f_{1},\ldots,f_{d},D\rangle$ the Northcott ideal associated to $\Phi$ and $\mathbf{m}$ (see, for example, [27, p. 100]).”

Returning to our problem of interest, one can observe that, by taking

[TABLE]

and $\mathbf{m}=(x_{1}^{u_{21}},x_{2}^{u_{32}},x_{3}^{u_{13}}),$ we have that $\Phi\,\mathbf{m}^{\top}=(f_{1},f_{2},f_{3})^{\top}$; $\Phi\,(x_{1}^{u_{21}},x_{2}^{u_{32}},x_{3}^{u_{13}})^{\top}=(x_{1}^{u_{41}+u_{21}}-x_{3}^{u_{13}}x_{4}^{u_{14}},x_{2}^{u_{42}+u_{32}}-x_{1}^{u_{21}}x_{4}^{x_{24}},x_{3}^{u_{43}+u_{13}}-x_{2}^{u_{32}}x_{4}^{u_{34}})$ and $D=\det(\Phi)=x_{4}^{u_{14}+u_{24}+u_{34}}-x_{1}^{u_{41}}x_{2}^{u_{42}}x_{3}^{u_{43}}$. In other words, a toric ideal generated by (N) can be viewed as a Northcott type ideal by Proposition 1.5.

Clearly, we can generalize this situation in the following way.

Definition 2.1**.**

Let

[TABLE]

and $\mathbf{m}=(x_{1}^{u_{21}},\ldots,x_{n-2}^{u_{n-1n-2}},x_{n-1}^{u_{1n-1}}),$ with $u_{ij}>0,$ for all $i,j$. If $(f_{1},\ldots,f_{n-1})^{\top}=\Phi\,\mathbf{m}^{\top}$ and $D=\det(\Phi),$ then the ideal of $\Bbbk[\mathbf{x}]$ generated by $\{f_{1},\ldots,f_{n-1},D\}$ will be called a critical ideal of Northcott type.

Let us see that every critical ideal of Northcott type is the ideal associated to a (non-necessarily torsion-free) monoid.

Theorem 2.2**.**

If $J$ is a critical ideal of Northcott type, there exists a commutative finite group $T$ and $\mathcal{A}\subset T\oplus\mathbb{Z}$ such that $I_{\mathcal{A}}=J$. In particular, $\mathbb{Z}\mathcal{A}$ has rank one.

Proof.

To each binomial $\mathbf{x}^{\mathbf{v}}-\mathbf{x}^{\mathbf{v}^{\prime}}\in\Bbbk[\mathbf{x}]$ we associate the vector $\mathbf{v}-\mathbf{v}\in\mathbb{Z}^{n}$. With this notation, the rows of the following matrix

[TABLE]

correspond to the exponent vectors of the binomials $f_{1},\ldots,f_{n-1}$ and $D$ in Definition 2.1.

Since the last row of $M$ is the sum of the rows above and the minor obtained by removing the last row and column of $M$ is

[TABLE]

we have that $M$ has rank $n-1$. Therefore, there exist unimodular matrices $P$ and $Q$ such that $P^{-1}MQ=\Delta$ where $\Delta$ is the Smith normal form of $M$, that is, $\Delta$ is a diagonal matrix with entries in the diagonal equal to $d_{1},\ldots,d_{n}$ where $d_{1}\mid d_{2}\mid\cdots\mid d_{n-1}$ are positive integers and $d_{n}=0$. Now, it is well-known that if $i$ is the smallest index such that $d_{i}\neq 1$ and $A$ is the $(n-i+1)\times n$ matrix whose rows are the last $n-i$ columns of $Q$, then we can choose $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset\mathbb{Z}_{d_{i+1}}\oplus\cdots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{Z}$ with $\mathbf{a}_{i}$ the $i$th column of $A$. With this notation, the rows of $M$ generate the subgroup of $\mathbb{Z}^{n}$ defined as $\mathcal{M}=\{\mathbf{v}-\mathbf{v}^{\prime}\mid A\mathbf{v}=A\mathbf{v}^{\prime}\}$ (this is a standard procedure, see for instance [24, Chapter 3]).

Let us see that $\mathbb{N}\mathcal{A}$ is unit free. If there is $\mathbf{a},\mathbf{b}\in\mathbb{N}\mathcal{A}$ such that $\mathbf{a}+\mathbf{b}=0$, then there exists $\mathbf{v},\mathbf{v}^{\prime}\in\mathbb{N}^{n}$ such that $\mathbf{a}=A\mathbf{v}$ and $\mathbf{b}=A\mathbf{v}^{\prime}$. Hence $A(\mathbf{v}+\mathbf{v}^{\prime})=0=A0$, which in particular implies that $\mathbf{v}+\mathbf{v}^{\prime}\in\mathcal{M}\cap\mathbb{N}^{n}$. So, in order to prove that $\mathbb{N}\mathcal{A}$ is unit free, it suffices to show that $\mathcal{M}\cap\mathbb{N}^{n}=\{0\}$. And, as the last row of $M$ is the sum of the others, this is equivalent to prove that $(\alpha_{1},\ldots,\alpha_{n-1},0)M\in\mathbb{N}^{n}$ implies $\alpha_{i}=0$ for all $i$. The condition $(\alpha_{1},\ldots,\alpha_{n-1},0)M\in\mathbb{N}^{n}$ is defined by the following system of inequalities:

[TABLE]

If $\alpha_{i}\neq 0$ for some $i$, from the last inequality we obtain that $\alpha_{j}<0$ for some $j$. For simplicity, we suppose $j=1$. Thus, from the first inequality follows that $\alpha_{2}<\alpha_{1}$. By repeating this argument with the second inequality, the third inequality, …, and the $(n-1)$th inequality, we conclude that $\alpha_{1}<\alpha_{n-1}<\cdots<\alpha_{2}<\alpha_{1}<0$ which is impossible. Therefore $\alpha_{i}=0$ for every $i$.

With this we have that $\mathbb{N}\mathcal{A}$ is a unit free cancellative monoid, and that $J\subseteq I_{\mathcal{A}}$.

Now, as $\mathbb{N}\mathcal{A}$ is unit free, by [24, Proposition 3.10], we may suppose that $\mathcal{A}\subset\mathbb{Z}_{d_{i+1}}\oplus\cdots\oplus\mathbb{Z}_{d_{n}}\oplus\mathbb{N}$. Let $\mathbf{w}=(w_{1},\ldots,w_{n})\in\mathbb{N}^{n}$ be last row of $A$. Clearly the ideal $I_{\mathcal{A}}$ is homogeneous with respect to $\mathbf{w}$.

Let $\prec$ be the monomial order on $\Bbbk[\mathbf{x}]$ represented by a matrix whose first row is $\mathbf{w}$ and second row is $(0,\ldots,0,-1)$. We claim that $\mathcal{G}=\{f_{1},\ldots,f_{n-1},D\}$ is a Gröbner basis with respect to $\prec$.

Since \gcd\big{(}\mathrm{in}_{\prec}(f_{i}),\mathrm{in}_{\prec}(f_{j})\big{)}=1, it suffices to check that the $S$-polynomial $S(f_{i},D)$ reduces to zero with respect to $\mathcal{G}$, for every $i\in\{1,\ldots,n-1\}$. By direct computation, one can see that

[TABLE]

for every $i\in\{1,\ldots,n-1\}$, where $\sigma$ is the permutation $n-1\to n-2\to\cdots\to 2\to 1\to n-1$. Therefore our claim follows and, by [2, Theorem 3.1], we have that $\{f_{1},\ldots,f_{n-1},D\}$ is a Gröbner basis of $(J:x_{n}^{\infty})$, and consequently $(J:x_{n}^{\infty})=J$. Let $i\in\{1,\ldots,n-1\}$ and let $s_{i}$ is the smallest integer such that $(J:x_{i}^{s_{i}})=(J:x_{i}^{k}),$ for some (every) $k\geq s_{i}$; clearly, $s_{i}$ exists by the Noetherian property of $\Bbbk[\mathbf{x}]$. In this case, one has that $J=(J:x_{i}^{s_{i}})\cap(J+\langle x_{i}^{s_{i}}\rangle)$ (see, for instance [8, Proposition  7.2(a)]). Moreover, taking into account that $D=x_{1}^{u_{n1}}\cdots x_{n-1}^{u_{n\,n-1}}-x_{n}^{\sum_{i=1}^{n-1}u_{in}}\in J$, we have that $x_{i}^{s_{i}-u_{ni}}x_{n}^{\sum_{i=1}^{n-1}u_{in}}\in J+\langle x_{i}^{s_{i}}\rangle$ if $s_{i}\geq u_{ni}$ or $x_{n}^{\sum_{i=1}^{n-1}u_{in}}\in J+\langle x_{i}^{s_{i}}\rangle$ if $s_{i}<u_{ni}$. Therefore, putting all this together, either

[TABLE]

in contradiction with the minimality of $s_{i}$, or

[TABLE]

Therefore, we have that $J=((\ldots(J:x_{1}^{s_{1}}):\ldots):x_{n}^{s_{n}})=(J:(x_{1}\cdots x_{n})^{s}),$ for some $s$. And, by [8, Corollary 2.5], we conclude that $(J:(x_{1}\cdots x_{n})^{\infty})=I_{\mathcal{A}}$, whence $J=I_{\mathcal{A}}$.

The last statement is trivial. ∎

Definition 2.3**.**

Let $\mathcal{A}\subseteq\mathbb{Z}\otimes T$ where $T$ denotes a commutative finite group. We will say that $\mathbb{N}\mathcal{A}$ is a monoid of Northcott type if $I_{\mathcal{A}}$ is a critical ideal of Northcott type.

Observe that Theorem 2.2 establishes a bijective correspondence between monoid of Northcott type and critical ideal of Northcott type.

Recall that a monoid $\mathbb{N}\mathcal{A}$ is uniquely presented if $I_{\mathcal{A}}$ admits a unique minimal generating set of binomials (up to signs). Uniquely presented finitely generated commutative monoids where introduced in [11], as the semigroup counterpart of the toric ideals generated by indispensable binomials (see [21]).

Corollary 2.4**.**

Semigroups of Northcott type are uniquely presented.

Proof.

Let us see that every critical ideal of Northcott type has a unique minimal system of binomial generators (up to signs). By Theorem 2.2, there exist a finite subset of a group, $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\},$ such that $\mathbb{Z}\mathcal{A}$ has rank $1$ and $I_{\mathcal{A}}=\langle x_{i}^{c_{i}}-\mathbf{x}^{\mathbf{u}_{i}}\mid\mathbf{u}_{i}\in\mathbb{N}^{n}\!,i\in\{1,\ldots,n\}\rangle$, where $c_{i}\mathbf{e}_{i}-\mathbf{u}_{i}$ is the $i$th row of the matrix $M$ in the proof of Theorem 2.2. If $c_{i}\mathbf{a}_{i}\neq c_{j}\mathbf{a}_{j}$ for every $i\neq j$, by [11, Corollary 5], then $I_{\mathcal{A}}$ is uniquely generated and we are done. Otherwise, $\mathbf{x}^{\mathbf{u}_{i}}-\mathbf{x}^{\mathbf{u}_{j}}\in I_{\mathcal{A}}$ for some $i\neq j$. By construction, the supports of $\mathbf{u}_{i}$ and $\mathbf{u}_{j}$ are not disjoint. So, there exists $k$ and $\alpha_{k}\in\mathbb{N}$ such that $\mathbf{x}^{\mathbf{u}_{i}}-\mathbf{x}^{\mathbf{u}_{j}}=x_{k}^{\alpha_{k}}(\mathbf{x}^{\mathbf{u}^{\prime}_{i}}-\mathbf{x}^{\mathbf{u}^{\prime}_{j}})$. As $\mathbb{N}\mathcal{A}$ is cancellative, $\mathbf{x}^{\mathbf{u}^{\prime}_{i}}-\mathbf{x}^{\mathbf{u}^{\prime}_{j}}\in I_{\mathcal{A}}$. But this is a contradiction with $\{x_{i}^{c_{i}}-\mathbf{x}^{\mathbf{u}_{i}}\mid\mathbf{u}_{i}\in\mathbb{N}^{n}\!,i\in\{1,\ldots,n\}\}$ being a Gröbner basis, as it was demostrated in the proof of Theorem 2.2. ∎

On the primary decomposition of critical ideals of Northcott type

Let $\Phi$ and $\mathbf{m}$ be as in Definition 2.1, and let $J$ be the corresponding critical ideal of Northcott type. By Theorem 2.2, there exists a subset $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}$ of a finitely generated commutative group such that $\mathbb{Z}\mathcal{A}$ has rank $1$ and $J=I_{\mathcal{A}}$.

Let $\mathcal{M}$ be the subgroup of $\mathbb{Z}^{n}$ generated by the columns of the matrix $M$ in the proof of Theorem 2.2 and $\rho:\mathcal{M}\to\Bbbk^{*}$ is the group homomorphism such that $\rho(\mathbf{v})=1,$ for all $\mathbf{v}\in\mathcal{M}$. Then

[TABLE]

where $\mathbf{v}_{+}$ and $\mathbf{v}_{-}\in\mathbb{N}^{n}$ denote the positive part and negative part of $\mathbf{v}$, respectively.

Now, by simply studying the group structure of the finite group $\mathbb{Z}^{n}/\mathcal{M}$, one can explicitly describe the radical, the associated primes and a minimal primary decomposition of a lattice ideal using the method developed by [8] when $\Bbbk$ is algebraically closed.

We summarize the mentioned descriptions as in [19].

Definition 2.5**.**

If $\mathcal{L}$ is a subgroup of $\mathbb{Z}^{n},$ then the saturation of $\mathcal{L}$ is

[TABLE]

We say that $\mathcal{L}$ is saturated if $\mathcal{L}=\mathrm{Sat}(\mathcal{L}).$

Note that the group $\mathrm{Sat}(\mathcal{M})/\mathcal{M}$ is finite and $\mathrm{Sat}(\mathcal{M})\cong\mathbb{Z}$.

Proposition 2.6**.**

The ideal $I_{+}(\rho)$ is prime if and only if $\mathcal{M}$ is saturated.

Proof.

This is a consequence of Theorem 2.1(c) in [8]. ∎

Definition 2.7**.**

If $\mathcal{L}$ is a subgroup of $\mathbb{Z}^{n}$ and $p$ is a prime number, we define $\mathrm{Sat}_{p}(\mathcal{L})$ and ${\rm Sat^{\prime}}_{p}(\mathcal{L})$ to be the largest sublattices of $\mathrm{Sat}(\mathcal{L})$ containing $\mathcal{L}$ such that $\mathrm{Sat}_{p}(\mathcal{L})/\mathcal{L}$ has order a power of $p$ and $\mathrm{Sat^{\prime}}_{p}(\mathcal{L})/\mathcal{L}$ has order relatively prime to $p.$ If $p=0,$ we adopt the convention that $\mathrm{Sat}_{p}(\mathcal{L})=\mathcal{L}$ and ${\rm Sat^{\prime}}_{p}(\mathcal{L})=\mathrm{Sat}(\mathcal{L}).$

Theorem 2.8**.**

Assume $\Bbbk$ algebraically closed and $\mathrm{char}(\Bbbk)=p\geq 0.$ Write $g$ for the order of $\mathrm{Sat^{\prime}}_{p}(\mathcal{M})/\mathcal{M}.$ There are $g$ distinct group homomorphisms $\rho_{1},\ldots,\rho_{g}$ from $\mathrm{Sat^{\prime}}_{p}(\mathcal{M})$ to $\Bbbk^{*}$ extending $\rho$ and, for each $j$ a homomorphism $\rho^{\prime}_{j}$ from $\mathrm{Sat}(\mathcal{M})$ to $\Bbbk^{*}$ extending $\rho_{j}.$ There is a unique group homomorphism $\rho^{\prime}$ from $\mathrm{Sat}_{p}(\mathcal{M})$ to $\Bbbk^{*}$ extending $\rho.$ The radical, associated primes and minimal primary decomposition of $I_{\mathcal{A}}=I_{+}(\rho)$ are:

[TABLE]

[TABLE]

and

[TABLE]

where $I_{+}(\rho_{j})$ is $I_{+}(\rho^{\prime}_{j})$-primary. In particular, if $p=0,$ then $I_{+}(\rho)$ is a radical ideal. The associated primes $I_{+}(\rho^{\prime}_{j})$ of $I_{+}(\rho)$ are all minimal and have the same codimension $\mathrm{rank}(L_{\rho}).$

Proof.

Corollary 2.2 and 2.5 in [8]. ∎

In the light of the theorem above, it suffices to know the extensions of $\rho$ to $\mathrm{Sat}_{p}(\mathcal{M}),$ $\mathrm{Sat}(\mathcal{M})$ and $\ \mathrm{Sat^{\prime}}_{p}(\mathcal{M})$ to find the radical, associated primes and the minimal primary decomposition of a lattice ideal, respectively.

Example 2.9**.**

Let

[TABLE]

and $\mathbf{m}=(x_{1},x_{2}^{2},x_{3}).$ In this case,

[TABLE]

By computing the Smith normal form of $M$ (using GAP, for instance), we may take $\mathcal{A}=\{(4,14),(5,18),(4,13),(8,29)\}\subset\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}$.

Since $\mathbb{Z}^{4}/\mathcal{M}\cong\mathbb{Z}\otimes\mathbb{Z}/2\mathbb{Z}$ the corresponding critical ideal of Northcott type

[TABLE]

is not prime by Proposition 2.6. Moreover, by Theorem 2.8, we have that that $I_{\mathcal{A}}$ is radical and has two associated primes if $\mathrm{car}(\Bbbk)\neq 2$ and $J$ is primary if $\mathrm{car}(\Bbbk)=2$.

If $\mathrm{car}(\Bbbk)\neq 2$ the associated primes are minimally generated by

[TABLE]

Clearly, they are not critical ideals.

3. Numerical semigroups of Northcott type

In this section we focus on monoids of Northcott type that are numerical semigroups. Up to isomorphism, this means that the torsion part is equal to zero. We will study the invariants of the semigroup in terms of the corresponding exponents in the Northcott type ideal.

Let us see that is possible to find construct a subset $\mathcal{A}$ of $\mathbb{N}$ such that $\mathbb{N}\mathcal{A}$ is a numerical semigroup of Northcott type. Observe that a necessary and sufficient condition for $\mathcal{A}$ to be a system of generators of a numerical semigroup is that the nonzero $d_{i}$’s appearing in the proof of Theorem 2.2 are equal to $1$, and recall that $d_{i}$ is equal to greatest common divisor of the $i\times i$ minors of the matrix $M$ in the proof of Theorem 2.2.

Proposition 3.1**.**

Let $\Phi$ as in Definition 2.1 and $\mathbf{m}=(x_{1},\ldots,x_{n-1})$. With the same notation as in the proof of Theorem 2.2, we have $d_{1}=\dots=d_{n-2}=1$ and $d_{n-1}$ is equal to

[TABLE]

Proof.

In this case, the matrix $M$ equals

[TABLE]

Since the minor obtained by removing the two last rows and columns is equal to one, we have that $d_{n-2}=1$. We already know that $d_{n}=0$, because the last row of $M$ is the sum of the previous ones. So, it suffices to study the $(n-1)\times(n-1)$ minors of the first $n-1$ rows of $M$. Let $a_{i}$ be absolute value of the minor of $M$ obtained by removing the last row and the $i$th column of $M$. By direct computation, we have that

• •

$a_{n}=\prod_{i=1}^{n-1}(u_{ni}+1)-1$,

• •

a_{k}=u_{\sigma(k)n}+\big{(}\sum_{i=1}^{n-2}\prod_{j=1}^{i}(u_{n\,\sigma^{k}(j)}+1)\big{)}u_{\sigma^{k}(i+1)\,n}, for $k\in\{1,\ldots,n-1\}$,

where $\sigma$ is the permutation defined by the cycle $(1\cdots n-1)$. Now, since

[TABLE]

for all $k\in\{1,\ldots,n-1\}$, we easily derive $\gcd(a_{1},\ldots,a_{n-1},a_{n})=\gcd(a_{n-2},a_{n})$. ∎

Example 3.2**.**

By the above proposition, if we take $\mathbf{m}=(x_{1},\ldots,x_{n-1}),\ u_{nj}=1,\ j=1,\ldots,n-1,\ u_{in}=1,\ i=2,\ldots,n-1$ and $u_{1n}=2$, then the subsemigroup of $\mathbb{N}$ generated by

[TABLE]

is a numerical semigroup of Norhtcott type.

Let $S$ be a numerical semigroup, and let $z$ be an integer. The Apéry set of $z$ is the set

[TABLE]

It is well known that the cardinality of $\mathrm{Ap}(S,z)$ is greater than or equal to $z$ and that the equality holds if and only if $z\in S$ (see [25, Lemma 2.4]).

Lemma 3.3**.**

If $\mathbb{N}\mathcal{A}$ is a numerical semigroup of Northcott type, then

[TABLE]

Proof.

By the proof of Theorem 2.2, we have that $\mathcal{G}=\{f_{1},\ldots,f_{n-1},D\}$ is a Gröbner basis of $I_{\mathcal{A}}$ with respect to a monomial order on $\Bbbk[\mathbf{x}]$ represented by a matrix whose first row is $(a_{1},\ldots,a_{n})$ and second row is $(0,\ldots,0,-1)$. Now, our claim follows from the results in [14, Section 3] or more directly by [22, Theorem 3], by just taking into account that the initial ideal of $I_{\mathcal{A}}$, with respect to an $\mathcal{A}-$graded reverse lexicographical term order such that $x_{n}$ is the least variable, is spanned by $\{x_{1}^{u_{n1}+u_{21}},\ldots,x_{n-2}^{u_{n\,n-2}+u_{n-1\,n-2}},x_{n-1}^{u_{n\,n-1}+u_{1\,n-1}},x_{1}^{u_{n1}}\cdots x_{n-1}^{u_{n\,n-1}}\}$, as we have shown in the proof of Theorem 2.2. ∎

The Frobenius number of $S$ is the largest integer not in $S$, denoted by $\mathrm{F}(S)$. The genus of $S$, $\mathrm{g}(S)$, is the number of gaps of $S$, that is, the cardinality of $\mathbb{N}\setminus S$. These two invariants can be calculated from $\mathrm{Ap}(S,s)$ for $s\in S\setminus\{0\}$, by using Selmer’s formulas (see for instance [25, Proposition 2.12]):

1. (1)

$\mathrm{F}(S)=\max(\mathrm{Ap}(S,s))-s$, 2. (2)

$\mathrm{g}(S)=\frac{1}{s}\sum_{w\in\mathrm{Ap}(S,s)}w-\frac{s-1}{2}$.

The conductor of $S$ is the Frobenius number of $S$ plus one. We will denote it by $\mathrm{c}(S)$.

We say that an integer $z$ is a pseudo-Frobenius number if $z\not\in S$ and $z+s\in S$ for all $s\in S\setminus\{0\}$. We will denote by $\mathrm{PF}(S)$ the set of pseudo-Frobenius numbers of $S$, and its cardinality, is the type of $S$, denoted by $\mathrm{t}(S)$. It is known that the type of a numerical semigroup agrees with the Cohen-Macaulay type of $\Bbbk[\![S]\!]$.

Corollary 3.4**.**

If $\mathbb{N}\mathcal{A}$ is a numerical semigroup of Northcott type, $\sigma$ is the permutation determined by the cycle $(1\,2\,\cdots\,n-1)$ and N=\Big{(}\sum_{i=1}^{n-2}(u_{ni}+u_{i+1\,i}-1)a_{i}\Big{)}+(u_{n\,n-1}+u_{1\,n-1}-1)a_{n-1}, then

[TABLE]

In particular, $\mathrm{t}(\mathbb{N}\mathcal{A})=n-1$, and $\mathrm{F}(\mathbb{N}\mathcal{A})=N-\min\{u_{\sigma(j)\,j}a_{j}\mid j\in\{1,\ldots,n\}\}-a_{n}$. Moreover,

[TABLE]

Proof.

By [25, Proposition 2.20], we have that

[TABLE]

Now, by Lemma 3.3, our claim on the set of pseudo-Frobenius numbers follows. Notice that $N-u_{\sigma(j)\,j}a_{j}-a_{n}\neq N-u_{\sigma(k)\,k}a_{k}-a_{n}$ if $k\neq j$; otherwise $u_{\sigma(j)\,j}a_{j}=u_{\sigma(k)\,k}a_{k}$ in contradiction with the minimality of $c_{j}=u_{nj}+u_{\sigma(j)\,j}$.

From the definition of pseudo-Frobenius number we get that $\mathrm{F}(S)=\max(\mathrm{PF}(S))$, for any numerical semigroup $S$. Finally, the last formula is a consequence of Selmer’s formula for the genus of a numerical semigroup. ∎

In the case $\mathbf{m}=(x_{1},\ldots,x_{n-1})$, we can simplify the formulae in Corollary 3.4 as follows:

[TABLE]

and

[TABLE]

The first formula is a consequence of the equalities N=\sum_{i=1}^{n-1}u_{ni}a_{i}=\big{(}\sum_{i=1}^{n-1}u_{in}\big{)}a_{n} (notice that $\sum_{i=1}^{n-1}u_{in}$ is the last entry of the last row in the matrix $M$). The proof of the formula for the genus requires some extra computations:

[TABLE]

Wilf in [28] conjectured that for a numerical semigroup $S$, $\mathrm{c}(S)\leq\mathrm{e}(S)\mathrm{n}(S)$, where $\mathrm{n}(S)$ stands for the number of elements in the semigroup less than $\mathrm{F}(S)$. We see next that Wilf’s conjecture holds for numerical semigroups of Northcott type when $\mathbf{m}=(x_{1},\ldots,x_{n})$.

Observe that $\mathrm{n}(S)=\mathrm{c}(S)-\mathrm{g}(S)=\mathrm{F}(S)+1-\mathrm{g}(S)$.

[TABLE]

As we have

[TABLE]

we deduce

[TABLE]

For this last inequality we have used that $u_{in}\geq 1$, and that not all of them can be equal to one.

If $n\geq 4$, $(n-1)\frac{(2n-5)a_{n}-1}{2}-a_{n}\geq 2\cdot\frac{a_{n}-1}{2}-a_{n}=-1$, and so we conclude $\mathrm{e}(\mathbb{N}\mathcal{A})\mathrm{n}(\mathbb{N}\mathcal{A})-\mathrm{c}(\mathbb{N}\mathcal{A})>-1$. Hence $\mathrm{c}(\mathbb{N}\mathcal{A})\leq\mathrm{e}(\mathbb{N}\mathcal{A})\mathrm{n}(\mathbb{N}\mathcal{A})$.

If $n=3$, $u_{in}=1$ for $i\in\{1,\ldots n-2\}$, $u_{n-1\,n}=2$, and $c_{i}=2$ for every $i\in\{1,\ldots n-1\}$. We obtain a unique semigroup: $\langle 3,4,5\rangle$, and Wilf’s conjecture holds for this semigroup.

Example 3.5**.**

Let $n=5$. With the same notation as in Proposition 3.1, if $u_{5i}=i$, $i\in\{1,\ldots,4\}$, $u_{15}=5$ and $u_{i5}=i$, $i\in\{2,\ldots,4\}$, then

[TABLE]

and $\mathcal{A}=\{359,199,139,123,119\}$. It is easy, from the last remark, to obtain

[TABLE]

and $\mathrm{g}(\mathbb{N}\mathcal{A})=(14-1)(119-1)/2=767$.

4. Nonunique factorization invariants for Northcott type semigroups

Some nonunique factorization invariants can be derived from a minimal presentation of the monoid (see for instance [10]). The maximum and minimum of the Delta set of the monoid and the catenary degree can be obtained once we know the elements in the monoid involved in a minimal presentation.

Let $G$ be a finitely generated commutative group, and let $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ be such that $\mathbb{N}\mathcal{A}$ is unit free. Recall that an element $\mathbf{b}\in\mathbb{N}\mathcal{A}$ is a Betti element of $\mathbb{N}\mathcal{A}$ if there exists $\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}$ with ${A}\mathbf{u}=\mathbf{b}$($=A\mathbf{v}$), where as usual $A$ denotes the matrix with columns the elements of $\mathcal{A}$.

For $\mathbf{s}\in\mathbb{N}\mathcal{A}$, the set of factorizations of $\mathbf{s}$ is the set

[TABLE]

As $\mathbb{N}\mathcal{A}$ is unit free, the set $\mathsf{Z}(\mathbf{s})$ has finitely many elements (this follows by Dickson’s Lemma, since they are all incomparable with respect to the usual partial order on $\mathbb{N}^{n}$). The monoid $\mathbb{N}\mathcal{A}$ is a unique factorization monoid if every element has a unique factorization. Notice that a semigroup of Northcott type is not a unique factorization monoid, since its Betti elements have at least two factorizations.

The length of a factorization $\mathbf{u}=(u_{1},\ldots,u_{n})$ of $\mathbf{s}$ is defined as $|\mathbf{u}|=u_{1}+\cdots+u_{n}$. The set of lengths of factorizations of $\mathbf{u}$ is thus defined as

[TABLE]

The monoid $\mathbb{N}\mathcal{A}$ is said to be half-factorial if for all $\mathbf{s}\in\mathbb{N}\mathcal{A}$ the set $\mathsf{L}(\mathbf{s})$ is a singleton. A way to measure how far a monoid is from being half-factorial is by means of the distances between consecutive factorization lengths. This is precisely the idea under the definition of Delta sets.

Since $\mathsf{L}(\mathbf{s})$ has finitely many elements, we can arrange its elements and write it as $\mathsf{L}(\mathbf{s})=\{l_{1}<\cdots<l_{t}\}$. The Delta set of $\mathbf{s}$ is then defined as

[TABLE]

The Delta set of $\mathbb{N}\mathcal{A}$ is

[TABLE]

Proposition 4.1**.**

Let $G$ be a finitely generated commutative group, and let $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ be such that $\mathbb{N}\mathcal{A}$ is of Northcott type. Let $u_{ij}$ be as in (1). Then

[TABLE]

and

[TABLE]

Proof.

By Corollary 2.4, $\mathbb{N}\mathcal{A}$ is uniquely presented. In particular, this means that every Betti element of $\mathbb{N}\mathcal{A}$ has exactly two different factorizations, [11]. These factorizations are precisely the exponents of the binomials $f_{1},\ldots,f_{n-1},D$. Hence for each Betti element its Delta set is just the length of the difference of the corresponding exponents. With this observation, the proof now follows from [5, Corollary 2.4, Theorem 2.5]. ∎

Now we see that the catenary degree of a semigroup of Northcott type can also be determined from the $u_{ij}$ in (1). The catenary degree measures how spread are the factorizations of the elements of a monoid.

Given two elements $\mathbf{u},\mathbf{v}\in\mathbb{N}^{n}$, define $\mathbf{u}\wedge\mathbf{v}$ as the minimum componentwise of $\mathbf{u}$ and $\mathbf{v}$, that is, the infimum in the lattice $\mathbb{N}^{n}$ endowed with the usual partial ordering. The distance between $\mathbf{u}$ and $\mathbf{v}$ is

[TABLE]

Let $N$ be a nonnegative integer, and let $\mathbf{s}\in\mathbb{N}\mathcal{A}$. For $\mathbf{u},\mathbf{v}\in\mathsf{Z}(\mathbf{s})$, an $N$-chain joining $\mathbf{u}$ and $\mathbf{v}$ is a sequence $\mathbf{u}^{1},\ldots,\mathbf{u}^{k}\in\mathsf{Z}(\mathbf{s})$ such that $\mathbf{u}=\mathbf{u}^{1}$, $\mathbf{v}=\mathbf{u}^{k}$, and $\mathrm{d}(\mathbf{u}^{i},\mathbf{u}^{i+1})\leq N$ for all $i\in\{1,\ldots,k-1\}$.

The catenary degree of $\mathbf{s}$, $\mathsf{c}(\mathbf{s})$, is the least positive integer $c$ such that for every two factorizations of $\mathbf{s}$, there exists a $c$-chain joining them. The catenary degree of $\mathbb{N}\mathcal{A}$ is the supremum of the catenary degrees of its elements.

Proposition 4.2**.**

Let $G$ be a finitely generated commutative group, and let $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ be such that $\mathbb{N}\mathcal{A}$ is of Northcott type. Let $u_{ij}$ be as in (1). Then

[TABLE]

Proof.

The proof goes as in Proposition 4.1. Since every Betti element has exactly two factorizations, its catenary degree is the maximum of their lengths. The proof follows by [6, Theorem 3.1]. ∎

5. Gluings and Northcott type ideals

In this section, we give a Northcott type description of the toric ideals generated by (gN1) and (gN2), and we give a method to produce critical binomial ideals for arbitrary $n$ that are not Northcott type in general.

Let $G$ be a finitely generated commutative group and let $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ be such that $\mathbb{N}\mathcal{A}\cap(-\mathbb{N}\mathcal{A})=\{0\}$. Assume that $\mathcal{A}=\mathcal{A}_{1}\cup\mathcal{A}_{2}$ with $\mathcal{A}_{1}\neq\varnothing\neq\mathcal{A}_{2}$. After reindexing the elements in $\mathcal{A}$ if necessary, we may suppose that $\mathcal{A}_{1}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{k}\}$ and $\mathcal{A}_{2}=\{\mathbf{a}_{k+1},\ldots,\mathbf{a}_{n}\}$. We say that $\mathbb{N}\mathcal{A}$ is a gluing of $\mathbb{N}\mathcal{A}_{1}$ and $\mathbb{N}\mathcal{A}_{2}$ if $I_{\mathcal{A}}$ has a system of generators of the form $B_{1}\cup B_{2}\cup\{\mathbf{x}^{\mathbf{u}}-\mathbf{x}^{\mathbf{v}}\}$, where $B_{1}\cup\{\mathbf{x}^{\mathbf{u}}\}\subseteq\Bbbk[x_{1},\ldots,x_{k}]$ and $B_{2}\cup\{\mathbf{x}^{\mathbf{v}}\}\subseteq\Bbbk[x_{k+1},\ldots,x_{n}]$. The concept gluing of semigroups was defined by Rosales in [23] for affine semigroups, but the results obtained in the first section of this paper can be applied in our context (see for instance [4]). We will only consider here proper gluings, that is, $\mathcal{A}$ will be a minimal generating system of $\mathbb{N}\mathcal{A}$.

It follows from the proof of [23, Theorem 1.4] that if $\mathbb{N}\mathcal{A}$ is the gluing of $\mathbb{N}\mathcal{A}_{1}$ and $\mathbb{N}\mathcal{A}_{2}$, and $\mathbf{u}$ and $\mathbf{v}$ are as in the preceding paragraph, then $\mathbb{Z}\mathcal{A}_{1}\cap\mathbb{Z}\mathcal{A}_{2}=d\mathbb{Z}$, where $d=\mathcal{A}\mathbf{u}=\mathcal{A}\mathbf{v}$.

If $\mathbb{N}\mathcal{A}$ is a numerical semigroup minimally generated by $\mathcal{A}$, and $d_{i}=\gcd\mathcal{A}_{i}$, $i\in\{1,2\}$, then $\mathbb{N}\mathcal{A}$ is a gluing of $\mathbb{N}\mathcal{A}_{1}$ and $\mathbb{N}\mathcal{A}_{2}$ if and only if $d_{1}\in\mathbb{N}\mathcal{A}_{2}\setminus\mathcal{A}_{2},\ d_{2}\in\mathbb{N}\mathcal{A}_{1}\setminus\mathcal{A}_{1}$ and $\gcd(d_{1},d_{2})=1$ (see [25, Chapter 8]).

In order to simplify the next statements, we will consider (gN1) as a subcase of (gN2) by taking $u_{12}=u_{32}=0$ and, consequently, $f_{1}=D$. Let us suppose now that $J$ is an ideal of $\Bbbk[\mathbf{x}]$ generated by (gN*).

Proposition 5.1**.**

Let $B$ be the matrix whose rows are the exponent vectors of $f_{1},f_{2}$ $D$, and $g$, that is,

[TABLE]

Notice that the first and the third rows might define the same binomial. Then there exists $\mathcal{A}\subset\mathbb{N}$ such that $J=I_{\mathcal{A}}$ if and only if

• (a)

$\{f_{1},f_{2},D\}$* generates a toric ideal in $\Bbbk[x_{1},x_{2},x_{3}]$.*

• (b)

*the *$(3,j)-$minors of $B$, $j\in\{1,\ldots,4\}$ are relatively prime.

Proof.

See [1, Propositions 3.3 and 3.4]. ∎

Corollary 5.2**.**

With the above notation, there exists $\mathcal{A}=\{a_{1},a_{2},a_{3},a_{4}\}\subset\mathbb{N}$ such that $J=I_{\mathcal{A}}$ if and only if $\mathbb{N}\mathcal{A}$ is a gluing of $c_{4}\mathbb{N}\mathcal{A}^{\prime}$ and $a_{4}\mathbb{N}$, with $\mathcal{A}^{\prime}=\{a_{i}/\mathrm{gcd}(a_{1},a_{2},a_{3})\mid i=1,2,3\}$.

Proof.

By [26, Proposition 4.13(a)], $I_{\mathcal{A}}\cap\Bbbk[x_{1},x_{2},x_{3}]$ is the toric ideal of $\mathcal{A}^{\prime}$. So, condition (a) in Proposition 5.1 implies that $I_{\mathcal{A}}=I_{\mathcal{A}^{\prime}}+\langle g\rangle$. Furthermore, by Proposition 5.1(b), $a_{i}=c_{4}a^{\prime}_{i}$, $i\in\{1,2,3\}$, and $c_{4}a_{4}=u_{41}a_{1}+u_{42}a_{2}+u_{43}a_{3}=c_{4}(u_{41}a^{\prime}_{1}+u_{42}a^{\prime}_{2}+u_{43}a^{\prime}_{3}),$ so we conclude that $\mathcal{A}$ is a gluing of $c_{4}\mathcal{A}^{\prime}\mathbb{N}$ and $a_{4}\mathbb{N}$. The converse is a direct consequence of Proposition 5.1 and the Herzog’s classification theorem of toric ideals associated to numerical semigroups minimally generated by three positive integers (see [12, Section 3], or [20, Theorem 2.2]). ∎

Let us see now that the gluing of a critical monoid with a free semigroup of rank one is again a critical monoid.

Proposition 5.3**.**

Let $G$ be a commutative group, $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ and $\mathbf{a}_{n+1}\in G$. If $\mathbb{N}\mathcal{A}$ is a critical monoid and $S=\mathbb{N}\mathcal{A}+\mathbb{N}\mathbf{a}_{n+1}$ is a gluing of $\mathbb{N}\mathcal{A}$ and $\mathbb{N}\mathbf{a}_{n+1}$, then $S$ is a critical monoid.

Proof.

We are assuming that $I_{\mathcal{A}}$ is critical, and so it is generated by a set of the form $\{x_{1}^{c_{1}}-\mathbf{x}^{\mathbf{u}_{1}},\ldots,x_{n}^{c_{n}}-\mathbf{x}^{\mathbf{u}_{n}}\}\subset\Bbbk[x_{1},\ldots,x_{n}]$, for some $\mathbf{u}_{1},\ldots,\mathbf{u}_{n}\in\mathbb{N}^{n}$, and where $c_{i}$ is the least multiple of $\mathbf{a}_{i}$ that belongs to $\mathbb{N}(\mathcal{A}\setminus\{\mathbf{a}_{i}\})$ (and consequently the $i^{th}$ coordinate of $\mathbf{u}_{i}$ is zero). On the other hand, by definition of gluing, the semigroup ideal of $S$, $I_{\mathcal{A}\cup\{\mathbf{a}_{n+1}\}}$, has a system of generators of the form $\mathcal{S}=\{x_{1}^{c_{1}}-\mathbf{x}^{\mathbf{u}_{1}},\ldots,x_{n}^{c_{n}}-\mathbf{x}^{\mathbf{u}_{n}}\}\cup\{x_{n+1}^{c_{n+1}}-\mathbf{x}^{\mathbf{u}_{n+1}}\}$, with $\mathbf{x}^{\mathbf{u}_{n+1}}\in\Bbbk[x_{1},\ldots,x_{n}]$ (recall that $I_{\mathbb{N}\mathbf{a}_{n+1}}=\langle 0\rangle$). Also, we know that $\mathbb{Z}\mathcal{A}\cap\mathbb{Z}\mathbf{a}_{n+1}=\mathbb{Z}(c_{n+1}\mathbf{a}_{n+1})$ (our definition of gluing implies that $S$ is minimally generated by $\mathcal{A}\cup\{\mathbf{a}_{n+1}\}$, and thus $c_{n+1}>1$).

So it only remains to prove that the $c_{i}$ are still the minimal positive integers such that $c_{i}\mathbf{a}_{i}\in\mathbb{N}((\mathcal{A}\cup\{\mathbf{a}_{n+1}\})\setminus\{\mathbf{a}_{i}\})$, and that $c_{n+1}$ is the minimum positive integer with $c_{n+1}\mathbf{a}_{n+1}\in\mathbb{N}\mathcal{A}$.

Let $i\in\{1,\ldots,n\}$ and $k$ a positive integer fulfilling that $k\mathbf{a}_{i}=\sum_{j=1}^{n}w_{j}\mathbf{a}_{j}+w_{n+1}\mathbf{a}_{n+1}$ for some $\mathbf{w}=(w_{1},\ldots,w_{n},w_{n+1})\in\mathbb{N}^{n+1}$ with $w_{i}=0$. Then $w_{n+1}\mathbf{a}_{n+1}\in\mathbb{Z}\mathcal{A}\cap\mathbb{Z}\mathbf{a}_{n+1}=\mathbb{Z}(c_{n+1}\mathbf{a}_{n+1})$. It follows that $c_{n+1}\mid w_{n+1}$, thus we may write $w_{n+1}=\ell c_{n+1}$. As $c_{n+1}\mathbf{a}_{n+1}=\sum_{i=1}^{n}u_{n+1\,i}\mathbf{a}_{i}$, we have that $(k-\ell u_{n+1\,i})\mathbf{a}_{i}=(w_{1}+\ell u_{n+1\,1})\mathbf{a}_{1}+\cdots+(w_{i-1}+\ell u_{n+1\,i-1})\mathbf{a}_{i-1}+(w_{i+1}+\ell u_{n+1\,i+1})\mathbf{a}_{i+1}+\cdots+(w_{n}+\ell u_{n+1\,n})\mathbf{a}_{n}$. Since $\mathbb{N}\mathcal{A}$ is free of units, we deduce that $k-\ell u_{n+1\,i}>0$. The minimality of $c_{i}$ implies that $c_{i}\leq k-\ell u_{n+1\,i}\leq k$.

Finally, if $k$ is a positive integer such that $k\mathbf{a}_{n+1}\in\mathbb{N}\mathcal{A}$, then $k\mathbf{a}_{n+1}\in\mathbb{Z}\mathcal{A}\cap\mathbb{Z}\mathbf{a}_{n+1}=\mathbb{Z}(c_{n+1}\mathbf{a}_{n+1})$. This forces $c_{n+1}\mid k$, whence $c_{n+1}\leq k$. ∎

Observe also that we can use Theorem 2.2 and Proposition 5.3 to produce a new critical monoid. In fact, we can repeat this process and gluing again with another copy of $\mathbb{N}$ and so on. However, it is convenient to point that, for $n>4$, there are numerical semigroups whose critical associated ideal is not of this form nor Northcott type.

Example 5.4**.**

A quick search using numericalsgps ([7], a GAP package, [9]), shows that $S=\left<11,13,14,15,19\right>$ is not a gluing and not Northcott type.

Ψgap> MinimalPresentationOfNumericalSemigroup(s); [ [ [ 0, 0, 0, 2, 0 ], [ 1, 0, 0, 0, 1 ] ], [ [ 0, 0, 2, 0, 0 ], [ 0, 1, 0, 1, 0 ] ], [ [ 0, 2, 0, 0, 0 ], [ 1, 0, 0, 1, 0 ] ], [ [ 1, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 2 ] ], [ [ 3, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 1 ] ] ] Ψ

Unfortunately in the process of applying Proposition 5.3, the uniquely presented property of the family (N) might be lost. This is due to the fact that $c_{n+1}\mathbf{a}_{n+1}$ can be chosen to have more than one factorization in terms of the elements of $\mathcal{A}$. We illustrate this with an example.

Example 5.5**.**

Let $S=\langle 3,5,7\rangle$ and take $a=2$ and $b=21$.

gap> s:=NumericalSemigroup(3,5,7); <Numerical semigroup with 3 generators> gap> FactorizationsElementWRTNumericalSemigroup(21,s); [ [ 7, 0, 0 ], [ 2, 3, 0 ], [ 3, 1, 1 ], [ 0, 0, 3 ] ] gap> t:=NumericalSemigroup(6,10,14,21); <Numerical semigroup with 4 generators> gap> AsGluingOfNumericalSemigroups(t); [ [ [ 6, 10, 14 ], [ 21 ] ] ] gap> IsUniquelyPresentedNumericalSemigroup(t); false

In [11, Theorem 13] we characterized gluings that are uniquely presented. If we apply this characterization to $S$ in Proposition 5.3 with the notations in its proof, then $S$ is uniquely presented if and only if $\mathbb{N}\mathcal{A}$ is uniquely presented and $\pm(c_{n+1}\mathbf{a}_{n+1}-c_{i}\mathbf{a}_{i})\not\in S$ for all $i\in\{1,\ldots,n\}$. It is not hard to show that this is equivalent to $\mathbb{N}\mathcal{A}$ being uniquely presented and $c_{n+1}\mathbf{a}_{n+1}$ having a unique expression in terms of $\mathbf{a}_{1},\ldots,\mathbf{a}_{n}$.

6. On Northcott type ideals and set-theoretic complete intersections

A classic problem in Commutative Algebra is to ask what the least number of polynomials are needed to determine the radical of a given ideal. This number is called the arithmetic rank of the ideal. Clearly, for a binomial ideal $I$, one has the following inequalities

[TABLE]

where $\mathrm{ht}$, $\mathrm{ara}$ and $\mu$ denote the height, the arithmetic rank and the cardinal of a minimal system of generators of $I$ with smallest cardinality, respectively.

Definition 6.1**.**

An ideal $I$ of $\Bbbk[x_{1},\ldots,x_{n}]$ is said to be:

• •

a complete intersection if $\mu(I)=\mathrm{ht}(I)$,

• •

an almost complete intersetion if if $\mu(I)=\mathrm{ht}(I)+1$,

• •

a set-theoretic complete intersection if $\mathrm{ara}(I)=\mathrm{ht}(I)$, that is to say, if there exists $h$ polynomials $g_{1},\ldots,g_{h}$ such that $\sqrt{I}=\sqrt{\langle g_{1},\ldots,g_{h}\rangle}$, where $h=\mathrm{ht}(I)$.

If $I\subset\Bbbk[x_{1},\ldots,x_{n}]$ is a critical ideal of Northcott type, by definition, there exist a matrix $\Phi$ and vector $\mathbf{m}$ such that $I$ is minimal generated by $\{f_{1},\ldots,f_{n-1},D\}$ where $(f_{1},\ldots,f_{n-1})^{\top}=\Phi\,\mathbf{m}^{\top}$ and $D=\det(\Phi)$. The ideal generated by $\{f_{1},\ldots,f_{n-1}\}$ is a complete intersection and $I$ is an almost complete intersection.

Proposition 6.2**.**

With the above notation,

[TABLE]

where $\langle\mathbf{m}\rangle$ is the monomial ideal generated by the entries of $\mathbf{m}$.

Proof.

By construction, it suffices to prove that $I\cap\langle\mathbf{m}\rangle\subseteq\langle f_{1},\ldots,f_{n-1}\rangle$. Let $f=\sum_{i=1}^{n-1}g_{i}f_{i}+gD\in I\cap\langle\mathbf{m}\rangle$. Since the first summand lies in $\langle f_{1},\ldots,f_{n-1}\rangle\subseteq I\cap\langle\mathbf{m}\rangle$, we need to prove that

[TABLE]

Moreover, since $\langle\mathbf{m}\rangle$ is a monomial ideal, $D=-x_{n}^{\sum_{i=1}^{n-1}u_{in}}+x_{1}^{u_{1n}}\cdots x_{n-1}^{u_{n-1\,n}}$ and no generator of $\langle\mathbf{m}\rangle$ depends on $x_{n}$ we may assume, without loss of generality, that $g$ is a minimal generator of $\langle\mathbf{m}\rangle$. Thus, let $g=x_{j}^{u_{\sigma(j)j}}$, where $\sigma=(1\,2\,\cdots n-1)\in S_{n}$. For the sake of simplicity suppose $j=1$, then

[TABLE]

and we are done. ∎

Since $\langle\mathbf{m}\rangle$ is a complete intersection, Proposition 6.2 has the following interpretation in the context of liaison theory: the affine variety of a critical ideal of Norhtcott type is directly CI-linked to the variety of $\langle\mathbf{m}\rangle$ (see [15, Definition 5.1.2]).

Corollary 6.3**.**

With the above notation,

[TABLE]

Proof.

By Proposition 6.2, $\langle f_{1},\ldots,f_{n-1}\rangle=I\cap\langle\mathbf{m}\rangle$. Now, since $\sqrt{I\cap\langle\mathbf{m}\rangle}=\sqrt{I}\cap\sqrt{\langle\mathbf{m}\rangle}$ and $\sqrt{\langle\mathbf{m}\rangle}=\langle x_{1},\ldots,x_{n-1}\rangle,$ our claim follows. ∎

Lemma 6.4**.**

With the above notation, the reduction of $f_{n-1}^{c_{1}\cdots c_{n-2}}$ modulo the ideal generated by $\{f_{1},\ldots,f_{n-2}\}$ is

[TABLE]

where $\sigma=(1\,2\,\ldots n-1)\in S_{n}$ and $c_{i}=u_{ni}+u_{i+1\,i}$, $i\in\{1,\ldots,n-2\}$.

Proof.

Set $c_{n-1}=u_{n\,n-1}+u_{1\,n-1}$ and $N=c_{1}\cdots c_{n-2}$. Then

[TABLE]

Let $S$ be the semigroup associated to the critical ideal of Northcott type generated by $\{f_{1},\ldots,f_{n-1},D\}$ and let $\prec$ be an $S-$graded reverse lexicographical order on $\Bbbk[x_{1},\ldots,x_{n}]$ such that $x_{1}\succ x_{2}\succ\ldots\succ x_{n-2}\succ x_{n}\succ x_{n-1}$. Since $f_{i}$, $i\in\{1,\ldots,n-1\}$, are $S-$homogeneous, $\{f_{1},\ldots,f_{n-2}\}$ forms a Gröbner basis with respect $\prec$. Now, by performing division by $\{f_{n-2},\ldots,f_{1}\}$, whenever $j\,u_{j\,u_{n\,n-1}}\geq c_{2}$, we may substitute $x_{n-2}^{j\,u_{n-1\,n-2}}$ by a monomial that involves $x_{n},x_{n-1},x_{n-2}$ and $x_{n-3}$, such that the exponent of $x_{2}$ is less than $c_{2}$. If we repeat this process with $x_{n-3},x_{n-4},\ldots$ we obtain an equivalent expression, $g$, of $f_{n-1}^{N}$ modulo $\langle f_{1},\ldots,f_{n-2}\rangle$ such that every monomial is multiple of $x_{n-1}$; in particular, the leading term of $g$ is $x_{n-1}^{\prod_{i=1}^{n-1}u_{\sigma(j)\,j}}x_{n}^{r}$ and its tail lies in $\langle x_{1},\ldots,x_{n-1}\rangle$. A careful computation shows that $P:=\prod_{i=1}^{n-1}u_{\sigma(j)\,j}$ is strictly smaller than the power of $x_{n-1}$ in any monomial of $g-x_{n-1}^{P}$. So, we can write $g=x_{n-1}^{P}(x_{n}^{r}+h)$ for some $h\in\langle x_{1},\ldots,x_{n-1}\rangle$. ∎

The following result generalizes [18, Theorem 3.1].

Theorem 6.5**.**

Every critical ideal of Northcott type is a set-theoretic complete intersection.

Proof.

Let $I$ be a critical ideal of Northcott type. By construction, there exists a matrix $\Phi$ as in (1) and $\mathbf{m}=(x_{1}^{u_{21}},\ldots,x_{n-2}^{u_{n-1\,n-2}},x_{n-1}^{u_{1\,n-1}}),$ with $u_{ij}>0$ for all $i,j,$ such that $I$ is generated by $f_{1},\ldots,f_{n-1}$ and $D=\det(\Phi)$, where $(f_{1},\ldots,f_{n-1})^{\top}=\Phi\,\mathbf{m}^{\top}$. Moreover, by Theorem 2.2, there exists $\mathcal{A}\subset\mathbb{Z}\oplus T$, where $T$ is a finite commutative group, such that $\mathbb{N}\mathcal{A}\cap(-\mathbb{N}\mathcal{A})=\{0\}$ and $I=I_{\mathcal{A}}$. In particular, $(I:x_{i})=I$ for every $i$, as it is stated in the proof of Theorem 2.2.

By Lemma 6.4, there exists $f=x_{n}^{r}+h$, with $r\geq 1$ and $h\in\langle x_{1},\ldots,x_{n-1}\rangle$, such that $f_{n-1}\in\sqrt{\langle f_{1},\ldots,f_{n-2},f\rangle}$. Moreover, $f\in(\langle f_{1},\ldots f_{n-1}\rangle:x_{n-1}^{\infty})\subset(I:x_{n-1}^{\infty})=I$. Thus,

[TABLE]

Now,

[TABLE]

Therefore, by Corollary 6.3,

[TABLE]

which implies $D\in\sqrt{\langle f_{1},\ldots,f_{n-2},f\rangle}$ and we are done. ∎

We end this paper by showing that, alike the generalization of the family (N), the families (gN*) are as well set-theoretically complete intersections.

Proposition 6.6**.**

Let $G$ be a commutative group, $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ and $\mathbf{a}_{n+1}\in G$. If $I_{\mathcal{A}}$ is a set-theoretic complete intersection and $S=\mathbb{N}\mathcal{A}+\mathbb{N}\mathbf{a}_{n+1}$ is a gluing of $\mathbb{N}\mathcal{A}$ and $\mathbb{N}\mathbf{a}_{n+1}$, then $I_{S}$ is a set-theoretic complete intersection.

Proof.

Let $h=\mathrm{ht}(I_{\mathcal{A}})$. By hypothesis, there exist $f_{1},\ldots,f_{h}$ such that $\sqrt{f_{1},\ldots,f_{h}}=\sqrt{I_{\mathcal{A}}}$. By construction, $I_{S}=I_{\mathcal{A}}+\langle f\rangle$ and $\mathrm{ht}(I_{S})=h+1$. Putting this together, we obtain that

[TABLE]

and we are done. ∎

Corollary 6.7**.**

Let $G$ be a commutative group, $\mathcal{A}=\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\subset G$ and $\mathbf{a}_{n+1}\in G$. If $\mathbb{N}\mathcal{A}$ is a monoid of Northcott type and $S=\mathbb{N}\mathcal{A}+\mathbb{N}\mathbf{a}_{n+1}$ is a gluing of $\mathbb{N}\mathcal{A}$ and $\mathbb{N}\mathbf{a}_{n+1}$, then $I_{S}$ is a set-theoretic complete intersection.

Proof.

The proof is a straightforward consequence of Theorem 6.5 and Proposition 6.6. ∎