In this work, we extend Saia’s results on the characterization of Newton non-degenerate ideals to the context of ideals in \( \mathcal{O}_{X(S)}\) , where \( X(S)\) is an affine toric variety defined by the semigroup \( S\subset \mathbb{Z}^{n}_{+}\) . We explore the relationship between the integral closure of ideals and the Newton polyhedron. We introduce and characterize non-degenerate ideals, showing that their integral closure is generated by specific monomials related to the Newton polyhedron.

The search for elements and tools to describe equisingularity conditions in families of analytic varieties is one of the main questions in Singularity Theory. The theory of the integral closure of ideals and modules provides a handy tool for studying equisingularity problems. For instance, in [1] Teissier used the integral closure of the ideals to study the equisingularity of hypersurface germ families. Inspired by Teissier’s work Gaffney used the integral closure to study the equisingularity of families of complete intersection with isolated singularities (ICIS) [2].

Although essential, it is not easy to compute the integral closure of an ideal. However, there are some cases where this computation is possible. For example, if \( I\) is an ideal generated by monomials on \( \mathcal{O}_n\) (the ring of germs of analytic functions \( f: (\mathbb{C}^n,0) \to (\mathbb{C},0)\) ), the integral closure \( \overline{I}\) of \( I\) is generated by the monomials \( x^k = x^{k_1} \dots x^{k_n}\) , \( k\) belonging to the Newton polyhedron of \( I\) . This result was extended by Saia [3], thus characterizing the class of Newton non-degenerate ideals \( I \subset \mathcal{O}_n\) . In [4] Bivià-Ausina computes the integral closure of Newton non-degenerate submodules \( M \subset \mathcal{O}_n^p\) , \( p \geq 1\) . Therefore, the author characterize a class of submodules \( M \subset \mathcal{O}_n^p\) such that the integral closure \( \overline{M}\) of \( M\) is easily computable.

In this work we generalize the results presented by Saia in [3] to the case of Newton non-degenerate ideals in \( \mathcal{O}_{X(S)}\) , in which \( X(S)\) is the affine toric variety defined by the semigroup \( S \subset \mathbb{Z}^n_{+}\) .

This work is organized as follows. In Section 1, we introduce some definitions and properties related to affine toric varieties and Newton polyhedron defined in the \( cone(S)\) . In Section 2, we present a relationship between the integral closure of ideals in \( \mathcal{O}_{X(S)}\) and the Newton polyhedron of \( I\) , proving the following theorem:

In Section 3, we introduce the class of non-degenerate ideals of \( \mathcal{O}_{X(S)}\) and establish a relationship between the Newton polyhedron of the ideal \( I\) and \( C(\overline{I})\) , which is Newton polyhedron of the ideal generated by all monomials that belong to the integral closure of the ideal \( I\) .

In Section 4, we characterize these ideals by showing that their integral closure is generated by specific monomials related to the Newton polyhedron, thereby proving our main theorem:

For the convenience of the reader and to fix some notation we review some general facts in order to establish our results.

2.1 Toric varieties.

2.2 Newton polyhedra.

In this section we present the definition of Newton polyhedron of ideals of the ring \( \mathcal{O}_{X(S)}\) . For the definition of Newton polyhedron of a function \( f: X(S) \to \mathbb{C}\) we can refer to [7] for instance.

Definition 3

The Newton polyhedron determined by \( A \subset S\) , denoted by \( \Gamma_{+} (A)\) , is the convex hull of the set \( \{k+v \ : \ k \in A , v \in cone(S)\}\) . If \( v\in c\check{on}e(S) \) , we define

\[ \ell(v, \Gamma_{+} (A))=\min\{ \langle k,v\rangle \ : \ k \in \Gamma_{+} (A) \} \]

and

\[ \Delta(v, \Gamma_{+} (A))= \{k \in \Gamma_{+} (A): \langle k,v\rangle=\ell(v, \Gamma_{+} (A))\}. \]

A subset \( \Delta \subseteq \Gamma_{+} (A)\) is called a face of \( \Gamma_{+} (A)\) when there exists \( v \in c\check{on}e(S) \) such that \( \Delta= \Delta(v, \Gamma_{+} (A))\) .

Since a polyhedron can be described as the intersection of the half-planes defined by its faces, we have an important characterization.

Lemma 1

Let \( \Gamma_{+} (A) \subseteq cone(S)\) be the Newton polyhedron of a set \( A \subset S\) . Then

\[ \Gamma_{+}(A) = \{ x \in cone(S): \langle x,v \rangle \geq \ell(v, \Gamma_{+}(A)), for all v \in c\check{on}e(S) \}. \]

In the sequence, we define the Newton polyhedron of an ideal \( I\) of the ring \( \mathcal{O}_{X(S)} \) . First, let us introduce the Newton polyhedron of germs in \( \mathcal{O}_{X(S)} \) .

Definition 4

Let \( g \in \mathcal{O}_{X(S)} \) , i.e., \( g=f\vert_{X(S)}\) , where \( f: \mathbb{C} ^{r} \to \mathbb{C} \) is an analytic function such that \( f=\sum_{k} a_{k}x^{k}\) is its Taylor expansion. The support of \( g \in \mathcal{O}_{X(S)} \) is defined by the set

\[ supp (g) = \big\{k_1b_1+\cdots+k_rb_r \in S: a_k \neq 0 \text{ and }k=(k_1,\dots,k_r)\big\}, \]

where \( b_1,\dots, b_r\) are generators of \( S\) . The Newton polyhedron of \( g\) is defined by \( \Gamma_{+} (g)= \Gamma_{+} (supp(g))\) .

If \( v \in c\check{on}e(S) \) , we define \( \ell(v,g)=\ell(v, \Gamma_{+} (g))\) and \( \Delta(v,g)=\Delta(v, \Gamma_{+} (g))\) . Consider \( G \subseteq \mathcal{O}_{X(S)} \) . We denote by \( supp(G)\) the union of the supports of the elements of \( S\) and define the Newton polyhedron of \( G\) as \( \Gamma_{+} (G)= \Gamma_{+} (supp(G))\) . If \( I \subseteq \mathcal{O}_{X(S)} \) is an ideal generated by \( G\) , then \( \Gamma_{+} (I)= \Gamma_{+} (G)\) . That is, if \( G = \{g_{1},\dots,g_{p}\}\) is a generating system of \( I\) , then \( \Gamma_{+} (I)\) is equal to the convex hull of \( \Gamma_{+} (g_{1}) \cup …\cup \Gamma_{+} (g_{p})\) .

As in [4], we denote by \( I^{0}\) the ideal of \( \mathcal{O}_{X(S)} \) generated by the monomials \( x^{k}\) such that \( k_{1}b_{1}+…+k_{r}b_{r} \in \Gamma_{+} (I)\) , where \( x_1,\dots,x_r\) is a coordinate system in \( X(S)\) .

Let \( I\subseteq \mathcal{O}_{X(S)} \) be an ideal. An element \( h \in \mathcal{O}_{X(S)} \) is said to be integral over \( I\) if it satisfies an integral dependence relation

The set of elements \( h \in \mathcal{O}_{X(S)} \) which are integral over \( I\) forms an ideal of \( \mathcal{O}_{X(S)} \) , called the integral closure of \( I\) . We denote the integral closure of \( I\) by \( \overline{I}\) . If \( \overline{I} = I\) , then \( I\) is called integrally closed.

In [1] the following equivalences are proved.

Based in [1] and considering monomial ideals in \( \mathcal{O}_{X(S)} \) , we will show that if \( h \in \overline{I}\) , then all monomials that appear in \( h\) are represented in \( cone(S)\) by points located in \( \Gamma_{+} (I)\) .

Using this result, we can prove the following corollary.

In [8, Prop. 3.8] the authors also study the integral closure of monomial ideals in \( \mathcal{O}_{X(S)}\) . We remark that in our work the monomial hypothesis is not necessary to obtain the related results. According to item \( (ii)\) of Corollary 1, we can affirm that \( \overline{I} \subset I^0\) . However, it is not always true that \( I^0 \subset \overline{I}\) , as we can see in the following example.

Example 1

Let \( S\) be the semigroup generated by the set \( \{(1,0),(1,1),(1,2) \} \subset \mathbb{Z} ^{2}_{+}\) . Then, the toric surface \( X(S) = V(I_S) \subset {\mathbb{C}^3}\) , where \( I_S\) is the ideal generated by the binomial \( xz - y^2\) . Consider \( I \subset \mathcal{O}_{X(S)}\) the ideal generated by \( f=x^2y+3x^2z \) and \( g =y^3-xy+z^2\) . Then, \( supp(I)=\{(3,1),(3,2),(3,3),(2,1),(2,4)\},\) and the Newton polyhedron of \( I\) is represented in the Figure 2. Note that \( y^2 \in I^0\) , since \( (2,2) \in \Gamma_{+} (I)\) . But \( y^2 \notin \overline{I}\) , because there is an analytic curve \( \psi: ( \mathbb{C} ,0) \to (X(S),0)\) defined by \( \psi(t)=(t,t,t)\) such that \( y^2 \circ \psi\notin\psi^*(I)\mathcal{O}_1\) .

In this section, we present the concept of non-degenerate ideals and establish a relationship between the Newton polyhedron of the ideal \( K_{I}\) , which is generated by all monomials that belong to the integral closure of the ideal \( I\) , and the Newton polyhedron of \( I\) .

Let \( g \in \mathcal{O}_{X(S)} \) . As we said before, this means that \( g=f\vert_{X(S)}\) , where \( f: \mathbb{C} ^{r} \to \mathbb{C} \) is an analytic function, and then can be written as \( f=\sum_{k} a_{k}x^{k}\) . Now, given a compact subset \( A \subseteq cone(S)\) , we denote by

We set \( g_{A} = 0\) whenever \( A \cap supp(g) = \emptyset\) .

Let \( g\) be a representative of the germ \( g \in \mathcal{O}_{X(S)}\) such that \( 0 \notin supp(g)\) . Then, we associate to \( g\) a polynomial \( L(g) \in \mathcal{O}_{n} \) given by

where \( a_v=a_k\) , \( a_k\neq 0\) and \( v=k_1b_1+…+ k_rb_r.\)

In the definition above we ask only for algebraic conditions. This will be enough for our purposes. However, in the literature, we can find other definitions that also require geometrical conditions. For instance, in [7] Matsui and Takeuchi define the concept of Newton non-degenerate functions and complete intersection to study Milnor fiber over singular toric varieties. For this purpose, they also used geometrical conditions.

Yoshinaga proved in [9] that a function \( f\in \mathcal{O}_{n} \) is Newton non-degenerate if and only if the integral closure of the ideal

is generated by the monomials \( x^{k}\) such that \( k \in \Gamma_{+} (f)\) . Saia extended this result in [3], proving that an ideal \( I \subseteq \mathcal{O}_{n} \) is Newton non-degenerate if and only if \( \overline{I}\) is generated by the monomials \( x^{k}\) such that \( k \in \Gamma_{+} (I).\) In the following section, we extend this result to the ideals in \( \mathcal{O}_{X(S)} \) .

Observe that \( C(\overline{I})\) is the Newton polyhedron of the ideal \( K_I\) generated by all monomials belonging to the integral closure of \( I\) , i.e., \( K_I=\langle \{x^m: x^m \in \overline{I}\} \rangle\) . This is because \( \overline{I}\) is an ideal of \( \mathcal{O}_{X(S)}\) .

Our main goal is to prove that an ideal \( I \subseteq \mathcal{O}_{X(S)} \) is non-degenerate if and only if \( C(\overline{I})= \Gamma_{+} (I)\) . To do this, we will consider the construction of the toroidal embedding associated with the Newton polyhedron \( \Gamma_{+} (I)\) .

In [3], Saia generalized Yoshinaga’s result [9] using the construction of a toroidal embedding associated with the Newton polyhedron of a finite codimensional ideal \( I\) in \( \mathcal{O}_{n} \) to show that \( I\) is Newton non-degenerate if and only if the Newton polyhedron of \( I\) is the convex hull of the set of \( n\) -tuples \( m=(m_1,\dots,m_n)\) such that \( x^m \in \overline{I}\) . The procedure for constructing the toroidal embeddings associated with a given Newton polyhedron is a local modification of Khovanskii’s method of assigning a compact complex non-singular toroidal manifold to an integer-valued compact convex polyhedron in \( \mathbb{C} ^n\) . Here, we apply this procedure to the Newton polyhedron in the \( cone(S)\) . This construction is the essential tool used to prove our main result.

For any ideal \( I \subseteq \mathcal{O}_{X(S)}\) we describe a partition of \( c\check{on}e(S) \) into convex cones with respect to \( \Gamma_{+}(I)\) .

We define an equivalence relation on \( c\check{on}e(S) \) by

Considering this equivalence, any equivalence class is naturally identified with a convex cone with its vertex at zero, specified by finitely many linear equations and strictly linear inequalities with rational coefficients.

The closures of equivalence classes specify a partition \( \Sigma_{0}\) of \( c\check{on}e(S) \) into closed convex cones that have the properties:

• If \( \sigma_1\) is a face of a cone \( \sigma\) in \( \Sigma_0\) , then \( \sigma_{1} \in \Sigma_{0}\) .

• For any cones \( \sigma_1\) and \( \sigma_2\) in \( \Sigma_{0}\) , \( \sigma_{1} \cap \sigma_{2}\) is a face of both \( \sigma_{1}\) and \( \sigma_{2}\) .

In this way, based on \( \Sigma_{0}\) and using the algorithm described in the proof of Theorem \( 11\) from [10], we construct a partition \( \Sigma\) of \( c\check{on}e(S) \) into finitely many closed convex cones with their vertices at zero such that:

1. 1.

   Any cone belonging to \( \Sigma\) lies in one of the cones in \( \Sigma_{0}\) and is specified by finitely many linear equations and linear inequalities with rational coefficients.
2. 2.

   If \( \sigma_1\) is a face of a cone \( \sigma\) in \( \Sigma\) , then \( \sigma_{1} \in \Sigma\) .
3. 3.

   For any cones \( \sigma_1\) and \( \sigma_2\) in \( \Sigma\) , \( \sigma_{1} \cap \sigma_{2}\) is a face of both \( \sigma_{1}\) and \( \sigma_{2}\) .
4. 4.

   Any cone \( q\) -dimensional \( \sigma\) in \( \Sigma\) is simplicial and unimodular, i.e., there exists a set of primitive integer vectors \( a^{1}(\sigma),\dots,a^{q}(\sigma)\) which are linearly independent over \( \mathbb{R} \) and \( n-q\) primitive integer vectors \( a^{q+1}(\sigma),\dots,a^{n}(\sigma)\) such that

   \[ \mathbb{Z} a^{1}(\sigma)+ \cdots + \mathbb{Z} a^{n}(\sigma)= \mathbb{Z} ^{n}. \]

Based on these properties, we can make the following observations:

For a \( 1\) -dimensional cone \( \sigma\) in \( \Sigma\) , where \( a^{i}(\sigma)\) represents the corresponding primitive integer of \( \sigma\) , the set \( \Delta(a^{i}(\sigma))\) is a closed face of dimension \( (n-1)\) of \( \Gamma_{+} (I)\) .

For each compact face \( \Delta \) of \( \Gamma_{+} (I)\) , there exists a cone \( \sigma \in \Sigma\) of dimension \( n\) and a subset \( J\) of \( \{1,\dots,n\}\) such that \( \Delta\) can be expressed as the intersection \( \bigcap_{j \in J} \Delta(a^{n}(\sigma))\) , where \( a^{1}(\sigma),\dots,a^{n}(\sigma)\) is the corresponding set of primitive integer vectors of \( \sigma\) . We shall denote this face \( \Delta\) by \( \Delta_J\) . Notice that \( \Delta_{J}=\Delta\big(\sum_{i\in J}a^{i}(\sigma)\big)\) .

For all \( n\) -dimensional cones \( \sigma \in \Sigma\) , \( \bigcap^{n}_{i=1} \Delta(a^{i}(\sigma))\) is a one-point set.

Let \( \sigma\) be a \( n\) -dimensional cone in \( \Sigma\) and \( a^{1}(\sigma),\dots,a^{n}(\sigma)\) the corresponding set of primitive integer vectors of \( \sigma\) that has been ordered. We associate to each such \( \sigma\) a copy of \( \mathbb{C} ^{n}\) denoted by \( \mathbb{C} ^{n}(\sigma)\) . Let us denote by \( \pi_{\sigma}: \mathbb{C} ^{n}(\sigma)\to X(S)\) the mapping given by

where \( y_1,y_2,\dots,y_n\) are the coordinates in \( \mathbb{C} ^n(\sigma)\) and \( a^{i}(\sigma)=(a^{i}_{1}(\sigma),\dots,a^{i}_{n}(\sigma))\) .

Clearly, \( \pi_{\sigma}\) is a surjective map, since it is the composition \( \gamma \circ \Omega_{\sigma}\) of the following surjective applications

and

where \( b_1,\dots,b_r\) are the generators of \( S\) .

Consider the disjoint union \( \displaystyle\mathcal{C}= \bigcup_{\sigma \in \Sigma} \mathbb{C} ^{n}(\sigma)\) . We define the following equivalence relation: given two points \( y_{\sigma} \in \mathbb{C} ^{n}(\sigma)\) and \( y_{\tau} \in \mathbb{C} ^{n}(\tau)\) ,

We will denote this set thus obtained by \( X=X(\Gamma_{+}(I))= \mathcal{C}/\sim\) .

Since \( \Sigma\) satisfies properties \( 1-4\) , by Theorems \( 6\) , \( 7\) and \( 8\) of [10], we have that \( X\) is a non-singular \( n\) -dimensional algebraic complex manifold and \( \pi: X \to X(S)\) defined by \( \pi(y)=\pi_{\sigma}(y_\sigma)\) is a proper analytic mapping onto \( X(S)\) , where \( y_{\sigma} \in \mathbb{C} ^{n}(\sigma)\) is a representative of the equivalence class \( y \in X\) .

Let \( J\) be a subset of \( \{1,\dots,n\}\) . Define

where \( y_\sigma =(y_{\sigma,1}, \dots, y_{\sigma,n})\) .

Based in [11, Lemma 3.1], we have the following lemma.

Consequently, \( y_{\sigma} \in \pi_{\sigma}^{-1}(0)\) if and only if there exists a compact face \( \Delta_J\) such that \( y_{\sigma} \in E_{\sigma,J}\) .

For each \( n\) -dimensional cone \( \sigma \in \Sigma\) and any generator \( g_i\) of \( I\) , let \( h_i(y_{\sigma})\) be the analytic germ of function given by

where \( g= \sum_{k}a_k x^k\) , \( a_v=a_k\) and \( v=k_1b_1+…+k_rb_r\) . The following equality is satisfied by \( h_i(y_{\sigma})\) :

Furthermore, if \( y_{\sigma} \in E_{\sigma,J}\) then

Therefore, the next proposition gives us a necessary and sufficient condition so that a monomial \( x^m\) belongs to the integral closure of the ideal \( I\subseteq \mathcal{O}_{X(S)} \) .

Now we present our main result, characterizing the ideals in \( \mathcal{O}_{X(S)}\) whose integral closure is generated by monomials with exponents that belong to the Newton polyhedron defined in \( \text{cone} (S) \subseteq \mathbb{R}^{n}_{+} \) , where \( S\) is a finitely generated semigroup in \( \mathbb{Z}^{n}_{+}\) .

Example 2

Let \( S=\langle (1,0),(1,1),(1,2) \rangle\) . Consider \( I \subset \mathcal{O}_{X(S)}\) the ideal generated by \( f=x^4y+z^2 \) and \( g=xyz+x^2y^3\) . Then, \( supp(I)=\{(5,1),(2,4),(3,3),(5,3)\}.\) Thus, the Newton polyhedron of \( I\) is represented in the Figure 3

We observe that \( I\) is non-degenerate. Then, we have that the integral closure of \( I\) is generated by the monomials \( x^{k}\) such that \( k_{1}b_{1}+…+k_{r}b_{r} \in \Gamma_{+} (I)\) , by Theorem 2.

The authors are very grateful to Carles Bivià-Ausina, who in one of his visits to São Carlos gave us the idea of studying this kind of problem using the combinatory from the toric varieties and semigroups.

Amanda S. Araújo is supported by CAPES grant number 88887.827300/2023-00. Thaís M. Dalbelo is supported by FAPESP-Grant 2019/21181-0 and by CNPq grant 403959/2023-3. Thiago da Silva is funded by CAPES grant number 88887.909401/2023-00 and CAPES grant number 88887.897201/2023-00.

Amanda S. Araújo (Federal University of São Carlos)

Thaís M. Dalbelo (Federal University of São Carlos)

Thiago da Silva (Federal University of Espírito Santo)