Journal of Algebraic Combinatorics

, Volume 44, Issue 1, pp 99–117 | Cite as

Parity binomial edge ideals

Article
  • 167 Downloads

Abstract

Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gröbner bases and are radical if and only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.

Keywords

Binomial ideals Primary decomposition Mesoprimary decomposition Binomial edge ideals Markov bases 

Mathematics Subject Classification

Primary 05E40 Secondary 13P10 05C38 

1 Introduction

A binomial is a polynomial with at most two terms, and a binomial ideal is a polynomial ideal generated by binomials. Binomial ideals appear frequently in mathematics and also applications to statistics and biology. This paper is about decompositions of binomial ideals which appear, for instance, in understanding the implications of conditional independence statements [4, Chapter 3], steady states of chemical reaction networks [2, 17], or combinatorial game theory [15, 16].

Decomposition theory of binomial ideals started with Eisenbud and Sturmfels’ fundamental paper [5] which proves the existence of binomial primary decomposition over algebraically closed fields. It can be seen, however, that the field assumption is not strictly necessary: A mesoprimary decomposition captures all combinatorial features and exists over any given field [13]. Separating the arithmetical and combinatorial aspects of binomial ideals is important for applications where binomial primary decompositions over the complex numbers are often inadequate since they obscure combinatorics and prevent interpretations of the indeterminates as, say, probabilities or concentrations.

Actual primary decompositions have been computed almost exclusively of radical ideals. It is a general feature of (meso)primary decomposition that the embedded primes and components remain elusive. The partial decomposition of the Mayr-Meyer ideals by Swanson illustrates quite beautifully the mess one typically encounters when trying to determine components over embedded primes [19]. The minimal primes are often combinatorially fixed and thus much better behaved. For instance, for lattice basis ideals they are entirely determined by the indeterminates they contain [10]. More examples of interesting combinatorial descriptions of minimal primes of binomial ideals appear, for instance, in [8, 11, 14]. In practice, binomial (primary) decompositions can be found with computer algebra. For experimentation, we used and recommend the packages Binomials [12] and BinomialEdgeIdeals [20] in Macaulay2 [6].

This paper is about a class of ideals whose primary decomposition depends on the characteristic of the field and is in general different from the mesoprimary decomposition. We decompose these ideals using a new technique and hope to add to the toolbox for binomial decompositions. To define the key player, let G be a simple undirected graph on \(V(G)\) and with edge set E(G). Let \(\Bbbk \) be any field and denote by \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}] = \Bbbk [x_i,y_i : i\in V(G)]\) the polynomial ring in \(2|V(G)|\) indeterminates.

Definition 1.1

The parity binomial edge ideal ofG is
$$\begin{aligned} {\mathcal {I}}_{G} := \left\langle x_ix_j - y_iy_j : \{i,j\} \in E(G) \right\rangle \subseteq \Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]. \end{aligned}$$

Parity binomial edge ideals share a number of properties with binomial edge ideals [8], but the combinatorics is subtler. Various properties related to walks in G depend on whether the walk has even or odd length (and hence the name). If G is bipartite, then everything reduces to the results of [8] as follows.

Remark 1.2

Let G be bipartite on the vertex set \(V_1 \dot{\cup }V_2\). Consider the ring automorphism of \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\) which exchanges \(x_i\) and \(y_i\) if \(i\in V_1\) and leaves all remaining indeterminates invariant. Under this automorphism, \({\mathcal {I}}_{G}\) is the image of the binomial edge ideal of G.

Definition 1.1 was suggested by Rafael Villarreal at the MOCCA Conference 2014 in Levico Terme. He asked whether parity binomial edge ideals are radical. Theorem 5.5 combined with Remark 5.1 says that this is the case if and only if G is bipartite, or \(\mathrm{char}\,(\Bbbk ) \ne 2\). We compute the minimal primes of \({\mathcal {I}}_{G}\) in Sect. 4. In Proposition 5.4, we write \({\mathcal {I}}_{G}\) as an intersection of binomial ideals whose combinatorics is simpler, since then a short induction shows that, under the field assumption, all occurring intersections are radical (Theorem 5.5) and hence \({\mathcal {I}}_{G}\) is radical. In \(\mathrm{char}\,(\Bbbk ) = 2\), we determine a primary decomposition (Theorem 5.9), which turns out to be also a mesoprimary decomposition (Theorem 5.10).

Our determination of the minimal primes goes a route that is familiar from [14]. We first determine generators of the distinguished component \({\mathcal {I}}_{G}:(\prod _{i\in V(G)}x_iy_i)^\infty \) (that is, a Markov basis) in Sect. 2. Binomials b that appear in the Markov basis but are not themselves contained in \({\mathcal {I}}_{G}\) have the property that \(mb \in {\mathcal {I}}_{G}\) for some monomial m. This means that \({\mathcal {I}}_{G}:b\) contains the monomial m, and thus, some minimal primes of \({\mathcal {I}}_{G}\) contain the indeterminates that constitute m. In the case of parity binomial edge ideals, the witness monomial can be found inductively using walks (Lemma 2.4).

Just looking at Definition 1.1, one may hope that parity binomial edge ideals would deform to monomial edge ideals under the Gröbner deformation. This is not the case as already the simplest examples show, but nevertheless, the lexicographic Gröbner basis has combinatorial structure and we describe it completely in Sect. 3.

Shortly before first posting this paper on the arXiv, the authors became aware of [9]. That paper contains a different analysis of radicality of parity binomial edge ideals. In characteristic two, the parity binomial edge ideal \({\mathcal {I}}_{G}\) coincides with the ideal \(L_G\) defined there; thus, radicality is clarified by their Theorem 1.2 which here appears as Remark 5.1. If the characteristic of \(\Bbbk \) is not two, the linear transformation \(x_i \mapsto x_i - y_i\), \(y_i \mapsto x_i + y_i\) maps the parity binomial edge ideal to the permanental edge ideal \(\Pi _G\) defined in [9, Section 3]. Radicality of this ideal is clarified in their Corollary 3.3 by means of a Gröbner bases calculation. Our approach here is different and was developed completely independently. In particular, our proof of radicality cannot use the Gröbner basis by Remark 3.12. Additionally, we can clarify the separation of combinatorics and arithmetics of \({\mathcal {I}}_{G}\) independent of \(\mathrm{char}\,(\Bbbk )\) and determine its mesoprimary decomposition.

1.1 Conventions and notation

For \(n\in \mathbb {N}_{>0}\), let \([n] := \{1,\dots ,n\}\). All graphs here are finite and simple, that is, they have no loops or multiple edges. For any graph G, \(V(G)\) is the vertex set and E(G) is the edge set. For any \(S\subseteq V(G)\), G[S] is the induced subgraph on S and for a sequence of vertices \(P=(i_1,\ldots ,i_r)\in V(G)^r\), \(G[P]:=G[\{i_1,\ldots ,i_r\}]\). Throughout we assume that G is connected and in particular has no isolated vertices if \(|V(G)|\ge 2\). According to Definition 1.1, if a graph is not connected then the parity binomial edge ideals of the connected components live in polynomial rings on disjoint sets of indeterminates such that the problem reduces to connected graphs. Despite this assumption, non-connected graphs appear. Thus, for any graph H, let \(c(H)\) be the number of connected components, \(c_0(H)\) the number of bipartite connected components, and \(c_1(H)\) the number of connected components which contain an odd cycle. We freely identify ideals of sub-polynomial rings of \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\) with their images in \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\). Likewise, ideals of \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\) that do not use some of the indeterminates are considered ideals of the respective subrings. A binomial is pure difference if it equals the difference of two monomials.

2 Markov bases

Markov bases were first defined for toric ideals, but the definition extends easily to other lattice ideals. In this paper, by a Markov basis we mean generators of \({\mathcal {I}}_{G}:(\prod _{i\in V(G)}x_iy_i)^\infty \), which is compatible with the extended notions of Markov bases used in [4, Section 1.3] and [18, Section 2.1].

Definition 2.1

Let G be a graph. A (vw)-walk of length\(r-1\) is a sequence of vertices \(v = i_1, i_2 \dots ,i_r = w\) such that \(\{i_k,i_{k+1}\}\in E(G)\) for all \(k\in [r-1]\). The walk is odd (even) if its length is odd (even). A path is walk that uses no vertex twice. A cycle is a walk with \(v=w\). The interior of a (vw)-walk \(P=(i_1,\dots ,i_r)\) is the set \(\mathrm {int}(P)=\{i_1,\dots ,i_r\}\setminus \{v,w\}\).

Remark 2.2

In this paper, a cycle is only defined with a marked start and end vertex. Consequently, the interior of a cycle (in the usual graph theoretic sense) also depends on the choice of this vertex.

Convention 2.3

When no ambiguity can arise, for instance, because the vertices are explicitly enumerated, we call a (vw)-walk simply a walk.

Lemma 2.4

Let P be an (ij)-walk in G and for \(k\in \mathrm {int}(P)\), let \(t_k\in \{x_k,y_k\}\) arbitrary. If P is odd, then
$$\begin{aligned} (x_ix_j-y_iy_j)\prod _{k\in \mathrm {int}(P)}t_k\in {\mathcal {I}}_{G}. \end{aligned}$$
If P is even, then
$$\begin{aligned} (x_iy_j-x_iy_j)\prod _{k\in \mathrm {int}(P)}t_k\in {\mathcal {I}}_{G}. \end{aligned}$$

Proof

We prove the statement by induction on the length r of P. If \(r=1\), the statement is true by definition, thus assume that \(r>1\). If \(\mathrm {int}(P)=\emptyset \), then P is odd, i is adjacent to j, and the claim holds trivially. If \(\mathrm {int}(P)\ne \emptyset \), pick a vertex \(s\in \mathrm {int}(P)\). Consider first the case that P is an odd walk. Exchanging the roles of i and j if necessary, we can assume that the (is)-subwalk of P is odd and that the (sj)-subwalk is even. Using the induction hypothesis, the binomials corresponding to these walks are in \({\mathcal {I}}_{G}\). Now, if \(t_s=x_s\), then
$$\begin{aligned} x_ix_sx_j\prod _{k\in \mathrm {int}(P)\setminus s} t_k \equiv _{{\mathcal {I}}_{G}}y_iy_sx_j \prod _{k\in \mathrm {int}(P)\setminus s} t_k \equiv _{{\mathcal {I}}_{G}}y_ix_sy_j \prod _{k\in \mathrm {int}(P)\setminus s} t_k \end{aligned}$$
where we have first applied a binomial corresponding to the odd (is)-subwalk (which may traverse j) and then a binomial corresponding to the even (sj)-subwalk of P (which may traverse i). If \(t_s=y_s\), then we first apply the (sj)-walk and then the (is)-walk. The induction step for an even walk is similar and omitted. \(\square \)

Remark 2.5

Lemma 2.4 also holds for odd cycles in which case we get that monomial multiples of \(x_i^2-y_i^2\) are contained in \({\mathcal {I}}_{G}\) for any vertex i that is contained in the same connected component as an odd cycle.

Let \(\{i,j\}\in E(G)\) and denote Open image in new window where \(e_i\) is the standard unit vector in \(\mathbb {Z}^{V(G)}\) corresponding to \(i\in V(G)\). With this notation, the generator \(x_ix_j-y_iy_j\) has exponent vector \((m_{\{i,j\}},-m_{\{i,j\}})^T \in \mathbb {Z}^{2|V(G)|}\). The exponent vectors of generators of \({\mathcal {I}}_{G}\) generate a lattice
$$\begin{aligned} {\mathcal {L}}_G= \mathbb {Z}\left\{ \begin{pmatrix} m_e\\ -m_e \end{pmatrix} : e \in E(G) \right\} = \mathrm{im}_{\mathbb {Z}}\begin{pmatrix} A_G\\ -A_G \end{pmatrix} \subseteq \mathbb {Z}^{2|V(G)|}, \end{aligned}$$
where \(A_G\) is the incidence matrix of G. Consequently, \({\mathcal {L}}_G\) is the Lawrence lifting of \(\mathrm{im}_{\mathbb {Z}}(A_G) \subseteq \mathbb {Z}^n\). Recall that a Graver basis of a lattice is the unique minimal subset of the lattice such that each element of the lattice is a sign-consistent linear combination of elements of the Graver basis (see [3, Chapter 3] for Graver basics). A standard fact about Lawrence liftings is that the Graver basis of \(\mathrm{im}_{\mathbb {Z}}(A_G)\) can be lifted to a Graver basis of \({\mathcal {L}}_G\), which here equals the universal Gröbner basis and any minimal Markov basis of \({\mathcal {L}}_G\) [1, Proposition 1.1]. To determine the Graver basis of \(\mathrm{im}_{\mathbb {Z}}(A_G)\), let
$$\begin{aligned}&{\mathcal {M}}^{ odd }_G:=\{e_i+e_j: \text { there is an odd }(i,j)\text{-walk } \text{ in } G\}\\&{\mathcal {M}}^{ even }_G:=\{e_i-e_j: \text { there is an even }(i,j)\text{-walk } \text{ in } G\} \setminus \{0\}. \end{aligned}$$
Note in particular that if there is an odd (ii)-walk, then \(2\cdot e_i\in {\mathcal {M}}^{ odd }_G\).

Proposition 2.6

The Graver basis of \(\mathrm{im}_{\mathbb {Z}}(A_G)\) is \(\pm ({\mathcal {M}}^{ odd }_G\cup {\mathcal {M}}^{ even }_G)\).

Proof

According to Pottier’s termination criterion [3, Algorithm 3.3], it suffices to check that the sum of two elements of \(\pm ({\mathcal {M}}^{ odd }_G\cup {\mathcal {M}}^{ even }_G)\) can be reduced to zero sign-consistently. If there are no cancelations in the sum, for example if the two summands have disjoint support, the sum is reduced by either of the summands. Cancelation among elements \(e_{i_1} \pm e_{i_2}\) and \(e_{j_1} \pm e_{j_2}\) can only occur if \(|\{i_1,i_2,j_1,j_2\}| \le 3\). Without loss of generality assume \(i_2 = j_1\). Thus, if cancelation occurs, the sum of two proposed Graver elements must equal \(\pm (e_{i_1} \pm e_{j_2})\) and this is either zero or another element in \(\pm ({\mathcal {M}}^{ odd }_G\cup {\mathcal {M}}^{ even }_G)\) by concatenation of walks. \(\square \)

Proposition 2.6 shows that the minimal Markov, or equivalently Graver, basis of the ideal saturation \({\mathcal {J}}_{G}:={\mathcal {I}}_{G}:(\prod _{i\in V(G)}x_iy_i)^\infty \) at the coordinate hyperplanes consists of the following binomials:

Proposition 2.7

$$\begin{aligned} {\mathcal {J}}_{G}= & {} \left\langle x_ix_j-y_iy_j: \mathrm{there\,is\,an\,odd }(i,j){-\mathrm walk\,in }G \right\rangle \nonumber \\&\ +\left\langle x_iy_j-y_jx_i: \mathrm{there\,is\,an\,even }(i,j){-\mathrm walk\,in }G \right\rangle . \end{aligned}$$
(2.1)

Proof

This is Proposition 2.6 and [1, Proposition 1.1]. \(\square \)

Example 2.8

Due to the odd cycle in the graph G in Fig. 1, for all pairs (ij) of vertices with \(i\ne j\), both \(x_ix_j - y_iy_j\) and \(x_iy_j - x_jy_i\) are contained in \({\mathcal {J}}_{G}\). Hence, the ideal \({\mathcal {J}}_{G}\) has 15 generators for odd walks and 15 for even walks with disjoint endpoints. Since G is not bipartite, \(x_i^2-y_i^2\in {\mathcal {J}}_{G}\) for all \(i\in [6]\). In total, a minimal Markov basis of \({\mathcal {J}}_{G}\) consists of 36 generators.

Remark 2.9

If G is bipartite, the reachability of vertices with even or odd walks is determined by membership in the two groups of vertices. Consequently, for each spanning tree \(T\subseteq G\) we have \({\mathcal {J}}_{T}={\mathcal {J}}_{G}\). This is not true if G has an odd cycle.

3 A lexicographic Gröbner basis

For this section, an ordering of \(V(G)\) is necessary. Fix any labeling \(V(G)\cong [n]\), and let \(\succ \) be the lexicographic ordering on \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\) induced by \(x_{1}\succ \cdots \succ x_{n}\succ y_{1}\succ \cdots \succ y_{n}\). For \(i,j\in V(G)\) write \(i\succ j\) if \(x_{i}\succ x_{j}\). We now describe the lexicographic Gröbner basis of the parity binomial edge ideal. This Gröbner basis reduces the binomial of an (ij)-walk P by the binomial of an (ik)-walk or that of a (kj)-walk for a suitable \(k\in \mathrm {int}(P)\). For example, if P is odd and if \(i\succ j\succ k\), then a binomial of P can be reduced to zero by the binomials corresponding to two subwalks (ik) and (kj) in G[P], independently of their parity and of the chosen binomial of P. The following definition identifies the configurations that lead to irreducible walk binomials.

Definition 3.1

Let P be an odd (ij)-walk with \(i\succeq j\) and for \(k\in \mathrm {int}(P)\), let \(t_k\in \{x_k,y_k\}\) be arbitrary. The binomial
$$\begin{aligned} (x_ix_j-y_iy_j)\prod _{k\in \mathrm {int}(P)}t_k \end{aligned}$$
(3.1)
is reduced if for all \(k\in \mathrm {int}(P)\)
  • there is no odd (ij)-walk in \(G[P\setminus \{k\}]\),

  • \(k\succ j\),

  • if \(i\succ k \succ j\), then all (ik)-walks in G[P] are odd and \(t_k=y_k\), and

  • if \(k\succ i\succ j\), then \(t_k=y_k\).

Let P be an even (ij)-walk with \(i\succ j\), and for \(k\in \mathrm {int}(P)\), let \(t_k\in \{x_k,y_k\}\) be arbitrary. The binomial
$$\begin{aligned} (x_iy_j-y_ix_j)\prod _{k\in \mathrm {int}(P)}t_k \end{aligned}$$
(3.2)
is reduced if for all \(k\in \mathrm {int}(P)\)
  • there is no even (ij)-walk in \(G[P\setminus \{k\}]\),

  • if \(i\succ j\succ k\), then all (ik)-walks in G[P] are either odd and \(t_k=y_k\) or they are all even and \(t_k=x_k\),

  • if \(i\succ k \succ j\), then all (ik)-walks in G[P] are odd and \(t_k=y_k\), and

  • if \(k\succ i\succ j\), then \(t_k=y_k\).

The set of reduced binomials is written \({{\mathcal {G}}}_\succ (G)\).

Clearly, \(x_ix_j-y_iy_j\in {{\mathcal {G}}}_\succ (G)\) for every edge \(\{i,j\}\in E(G)\). We make the reduced binomials more explicit as follows.

Remark 3.2

Let \(i\succeq j\) and let P be an (ij)-walk in G with \(\mathrm {int}(P)=\{i_1,\dots ,i_r\}\). Assume that there exist variables \(t_{k}\in \{x_{k},y_{k}\}\), \(k\in \mathrm {int}(P)\), such that the respective binomial in equation (3.1) or (3.2) is reduced. The case distinction in Definition 3.1 fixes the value of \(t_k\) for any \(k\in \mathrm {int}(P)\) as follows. Let
$$\begin{aligned} P^x:=\{k\in \mathrm {int}(P): j\succ k\text { and there is an even }(i,k)\text{-walk } \text{ in } G[P]\} \end{aligned}$$
and \(P^y:=\mathrm {int}(P)\setminus P^x\). In particular, \(P^y=\mathrm {int}(P)\) if P is odd. Thus,
$$\begin{aligned} \prod _{k\in \mathrm {int}(P)}t_k=\prod _{k\in P^x}x_k\prod _{k\in P^y}y_k. \end{aligned}$$

Example 3.3

Let P be an even (ij)-walk with \(i\succ j\) such that there exists \(k\in \mathrm {int}(P)\) with \(j\succ k\) and such that there exists an even and an odd (ik)-walk in G[P]. If \(t_k=x_k\) in Eq. (3.2), then the binomial can be reduced by the binomial corresponding to the odd (ik)-walk. If \(t_k=y_k\), then the binomial can be reduced to zero by the binomial corresponding to the even (ik)-walk.

Example 3.4

Consider the even walk (4, 3, 1, 2, 3, 6) in the parity binomial edge ideal for Fig. 1. The binomial \((x_4x_6-y_4y_6)y_3y_2y_1\) is reduced, whereas the binomial \((x_4x_6-y_4y_6)y_3x_2y_1\) is not. In particular, reduced odd walks can have odd cycles. In the even (4, 5)-walk \(P=(4,3,1,2,3,6,5)\), there exists an even (4, 6)-subwalk and an odd (4, 6)-subwalk in G[P]. Since \(4\succ 6\), no choice of variables \(t_k\in \{x_k,y_k\}\) makes the binomial \((x_4y_5-y_4x_5)t_1t_2t_3t_4t_6\) a reduced binomial.

The first step is to see that reduced binomials have minimal leading terms among all binomials in Lemma 2.4 corresponding to walks, justifying their name.

Lemma 3.5

Let P be an (ij)-walk and \(t_k\in \{x_k,y_k\}\) for \(k\in \mathrm {int}(P)\) arbitrary. Then, \((x_ix_j-y_iy_j)\prod _{k\in \mathrm {int}(P)}t_k\) if P is odd and \((x_iy_j-y_ix_j)\prod _{k\in \mathrm {int}(P)}t_k\) if P is even, reduce to zero modulo \({{\mathcal {G}}}_\succ (G)\).

Fig. 1

A graph with an even walk, but no even path from 4 to 5. The interior of the walk (4, 3, 1, 2, 3, 6, 5) is \(\{\textit{1}, \textit{2}, \textit{3}, \textit{6}\}\)

Proof

This is an induction on the length of P. Assume that the binomial contradicts the first bullet in the respective definition of being reduced, then it is a monomial multiple of a binomial of a shorter walk which can be reduced to zero by induction. If the binomial fulfills the first bullet, then there exist some \(k\in \mathrm {int}(P)\) that violates one of the other properties in the definition. In this case, there exist two subwalks (ik) and (kj) whose binomials reduce the original binomial (see Example 3.3), and which are themselves reducible by the induction hypothesis. \(\square \)

We now state the main theorem of this section. Its proof is by Buchberger’s criterion and splits into a couple of lemmas.

Theorem 3.6

The set \({{\mathcal {G}}}_\succ (G)\) of reduced binomials is the reduced Gröbner basis of \({\mathcal {I}}_{G}\) with respect to \(\succ \).

Remark 3.7

The Gröbner basis in [8] looks similar, but for the original binomial edge ideals there are no binomials corresponding to \(k\in \mathrm {int}(P)\) with \(i \succ k \succ j\) in the Gröbner basis. The Gröbner basis there is also not a subset of our Gröbner basis. For example, all Gröbner elements in [8] which come from admissible (ij)-paths can be reduced to zero by odd moves from \({{\mathcal {G}}}_\succ (G)\) if there exists an odd (ik)-subwalk with \(j\succ k\).

For the reduction of s-polynomials, we use the following well-known fact.

Lemma 3.8

Let \(f,g\in \Bbbk [{{\mathbf {x}}}]\) and \(\succ \) a monomial ordering. If their leading monomials form a regular sequence, \(\mathrm {spol}(uf,vg)\) reduces to zero for all monomials \(u,v\in \Bbbk [{{\mathbf {x}}}]\).

Lemma 3.9

Let \(g_P\) and \(g_Q\) be reduced binomials corresponding to even walks P and Q. Then, \(\mathrm {spol}(g_P,g_Q)\) reduces to zero with respect to \({{\mathcal {G}}}_\succ (G)\).

Proof

Let P be an even \((p_1,p_2)\)-walk with \(p_1\succ p_2\) and Q be an even \((q_1,q_2)\)-walk with \(q_1\succ q_2\). By Remark 3.2, we write
$$\begin{aligned} g_P= & {} (x_{p_1}y_{p_2}-y_{p_1}x_{p_2})\cdot \prod _{i\in P^x}x_i\cdot \prod _{i\in P^y}y_i\\ g_Q= & {} (x_{q_1}y_{q_2}-y_{q_1}x_{q_2})\cdot \prod _{i\in Q^x}x_i\cdot \prod _{i\in Q^y}y_i. \end{aligned}$$
If \(|\{p_1,p_2,q_1,q_2\}| = 4\), then \(x_{p_1}y_{p_2}\) and \(x_{q_1}y_{q_2}\) are coprime and thus form a regular sequence. Lemma 3.8 gives this case. If \(\{p_1,p_2,q_1,q_2\} = \{q_1,q_2\}\), then the s-polynomial is zero.
The only interesting case is when P and Q have precisely one endpoint in common. First, let that common endpoint be \(v:=p_1=q_1\). Since \(p_1\not \in Q^x\), \(q_1\not \in P^x\), and since we can assume that \(q_2\succ p_2\), the s-polynomial is
$$\begin{aligned} (x_{q_2}y_{p_2}-y_{q_2}x_{p_2})\cdot y_v\cdot \prod _{i\in P^x\cup Q^x}x_i\prod _{i\in (P^y\cup Q^y)\setminus \{q_2,p_2\}}y_i. \end{aligned}$$
This binomial is a monomial multiple of the binomial obtained from the \((q_2,p_2)\)-walk which might traverse the vertex \(v=p_1=q_2\). Hence, the s-polynomial reduces to zero by Lemma 3.5. The case that \(p_2=q_2\) is similar and omitted. The last case is (without loss of generality) \(q_1\succ q_2=p_1\succ p_2\). In this case, \(x_{q_1}y_{q_2}\) and \(x_{p_1}y_{p_2}\) form a regular sequence and due to Lemma 3.8\(\mathrm {spol}(g_P,g_Q)\) reduces to zero. \(\square \)

Lemma 3.10

Let \(g_P\) and \(g_Q\) be reduced binomials corresponding to odd walks P and Q. Then, \(\mathrm {spol}(g_P,g_Q)\) reduces to zero with respect to \({{\mathcal {G}}}_\succ (G)\).

Proof

Assume that P is a \((p_1,p_2)\)-walk with \(p_1\succ p_2\) and Q is a \((q_1,q_2)\)-walk with \(q_1\succ q_2\). Without loss of generality, let \(p_1\succ q_1\). By Lemma 3.8, we can assume \(|\{p_1,p_2\}\cap \{q_1,q_2\}|\ge 1\). Clearly, if \(\{p_1,p_2\}=\{q_1,q_2\}\), then \(\mathrm {spol}(g_P,g_Q)=0\). In total assume that \(\{p_1,p_2\}\ne \{q_1,q_2\}\). Under this assumptions, in all remaining cases, the s-polynomial is a monomial multiple of the binomial corresponding to the even walk which arises from gluing P and Q along the vertex they have in common. \(\square \)

Lemma 3.11

Let \(g_P\) and \(g_Q\) be reduced binomials corresponding, respectively, to an odd walk P and an even walk Q. Then, \(\mathrm {spol}(g_P,g_Q)\) reduces to zero with respect to \({{\mathcal {G}}}_\succ (G)\).

Proof

Let P be an \((p_1,p_2)\)-walk with \(p_1\succeq p_2\) and Q be an even \((q_1,q_2)\)-walk with \(q_1\succ q_2\). By Remark 3.2, we write
$$\begin{aligned} g_P= & {} (x_{p_1}x_{p_2}-y_{p_1}y_{p_2})\prod _{i\in \mathrm {int}(P)}y_i,\\ g_Q= & {} (x_{q_1}y_{q_2}-y_{q_1}x_{q_2})\prod _{i\in Q^x}x_i\prod _{i\in Q^y}y_i. \end{aligned}$$
By Lemma 3.8, it suffices to consider the case that \(q_1\in \{p_1,p_2\}\). If \(p_1=q_1\), then
$$\begin{aligned} \mathrm {spol}(g_P,g_Q) = (x_{p_2}x_{q_2}-y_{q_2}y_{p_2})y_{p_1}\prod _{i\in \mathrm {int}(P)\setminus q_2}y_i\prod _{i\in Q^y}y_i\prod _{i\in Q^x\setminus p_2}x_i. \end{aligned}$$
This s-polynomial is a monomial multiple of the binomial corresponding to some \((p_2,q_2)\)-walk, traversing \(p_1=q_1\) if necessary. Thus, it reduces by Lemma 3.5. The case that \(p_2=q_1\) is similar and omitted. \(\square \)

Proof of Theorem 3.6

According to Lemma 3.9, Lemma 3.10 and Lemma 3.11, the set \({{\mathcal {G}}}_\succ (G)\) fulfills Buchberger’s criterion and hence is a Gröbner basis of \({\mathcal {I}}_{G}\). By construction, the elements of \({{\mathcal {G}}}_\succ (G)\) are reduced with respect to \(\succ \). \(\square \)

Theorem 3.6 implies in particular that parity binomial edge ideals of bipartite graphs are radical (which they must be by Remark 1.2). This, however, does not require the square-free initial ideal: If \(\mathrm{char}\,(\Bbbk )\ne 2\), then all parity binomial edge ideals are radical by Corollary 5.5.

Remark 3.12

The parity binomial edge ideal \({\mathcal {I}}_{K_3}\) of the 3-cycle \(K_3\) cannot have a square-free initial ideal with respect to any monomial order. This follows from the fact that \({\mathcal {I}}_{K_3}\) is not radical in \({\mathbb {F}}_2[{{\mathbf {x}}},{{\mathbf {y}}}]\) (see Remark 5.1). If \({\mathcal {I}}_{K_3}\) had a square-free Gröbner basis over some field \(\Bbbk \), its binomials must be pure difference (since the generators of \({\mathcal {I}}_{K_3}\) are pure difference). The pure difference property yields that this Gröbner basis would also be a square-free Gröbner basis over every other field, in particular, over \({\mathbb {F}}_2\).

4 Minimal primes

Generally, the minimal primes of a binomial ideal come in groups corresponding to the sets of indeterminates they contain. To start, we determine the minimal primes of \({\mathcal {I}}_{G}\) that contain no indeterminates, that is, the minimal primes of \({\mathcal {J}}_{G}\). They follow quickly from the next lemma, together with the results in [5, Section 2].

Lemma 4.1

Apart from zero rows, the Smith normal form of \(\begin{pmatrix} A_G\\ -A_G \end{pmatrix}\) is the diagonal matrix \(\mathrm{diag}\,(1,\dots ,1,2,\dots ,2)\) whose number of entries 1 is \(|V(G)|-c(G)\) and the number of entries 2 equals \(c_1(G)\).

Proof

See [7, Theorem 3.3]. \(\square \)

The following ideals are the building blocks for the primary decomposition of \({\mathcal {J}}_{G}\). For any connected graph G with an odd cycle, let
$$\begin{aligned} {\mathfrak {p}}^+(G) = \left\langle x_i+y_i: i\in V(G) \right\rangle \quad \text {and} \quad {\mathfrak {p}}^-(G) := \left\langle x_i-y_i: i\in V(G) \right\rangle . \end{aligned}$$

Proposition 4.2

Let G be a graph consisting of bipartite connected components \(B_1,\dots ,B_{c_0(G)}\) and non-bipartite connected components \(N_1,\dots ,N_{c_1(G)}\). If \(\mathrm{char}\,(\Bbbk ) \ne 2\), then \({\mathcal {J}}_{G}\) is radical, and its minimal primes are the \(2^{c_1(G)}\) ideals
$$\begin{aligned} \sum _{i=1}^{c_0(G)} {\mathcal {J}}_{B_i} + \sum _{i=1}^{c_1(G)} {{\mathfrak {p}}}^{\sigma _i}(N_i), \end{aligned}$$
where \(\sigma \) ranges over \(\{+,-\}^{c_1(G)}\). On the other hand, if \(\mathrm{char}\,(\Bbbk ) = 2\), then
$$\begin{aligned} {\mathcal {J}}_{G} = \sum _{i=1}^{c_0(G)} {\mathcal {J}}_{B_i} + \sum _{i=1}^{c_1(G)} {\mathcal {J}}_{N_i} \end{aligned}$$
is primary of multiplicity \(2^{c_1(G)}\) over the minimal prime \(\sum _{i=1}^{c_0(G)} {\mathcal {J}}_{B_i} + \sum _{i=1}^{c_1(G)} {{\mathfrak {p}}}^+(N_i)\).

Proof

Assume first that \(\Bbbk \) is algebraically closed. According to [5, Corollary 2.2], the primary decomposition \({\mathcal {J}}_{G}\) is determined by the saturations of the character that defines the lattice ideal \({\mathcal {J}}_{G}\). If a graph is disconnected, then its adjacency matrix has block structure according to the connected components. Therefore, it suffices to assume that G is connected. If G is bipartite, then Lemma 4.1 and [5, Corollary 2.2] imply that the lattice ideal \({\mathcal {J}}_{G}\) is prime. We are thus left with the case that G is connected and not bipartite.

Assume first that \(\mathrm{char}\,(\Bbbk ) \ne 2\). Lemma 4.1 and [5, Corollary 2.2] together show that \({\mathcal {J}}_{G}\) is radical and has two minimal primes. We show that these are \({\mathfrak {p}}^+(G)\) and \({\mathfrak {p}}^-(G)\). The first step is \({\mathcal {J}}_{G}\subseteq {\mathfrak {p}}^+(G)\) using Proposition 2.7. Let \(i, j\in V(G)\), then \(x_ix_j-y_iy_j=x_i\cdot (x_j+y_j)-y_j\cdot (x_i+y_i)\in {\mathfrak {p}}^+(G)\) and \(x_iy_j-x_jy_i=x_i\cdot (x_j+y_j)- x_j\cdot (x_i+y_i)\in {\mathfrak {p}}^+(G)\). Similarly, \({\mathcal {J}}_{G}\subseteq {\mathfrak {p}}^-(G)\). Now let \({{\mathfrak {p}}}\supseteq {\mathcal {J}}_{G}\) be a prime ideal. If \({{\mathfrak {p}}}\) contains \(x_i + y_i\) for all i, then it is either equal to \({\mathfrak {p}}^+(G)\) or not minimal over \({\mathcal {J}}_{G}\). If there exists a vertex i such that \(x_i + y_i \notin {{\mathfrak {p}}}\), then since G has an odd cycle and is connected, for any vertex j there are both an odd and an even (ij)-walk. Thus,
$$\begin{aligned} (x_i+y_i)\cdot (x_j-y_j) = x_ix_j - y_iy_j + x_jy_i - x_iy_j \in {{\mathfrak {p}}}. \end{aligned}$$
Since \({{\mathfrak {p}}}\) is prime, it contains \(x_j - y_j\) for each j and thus \({\mathfrak {p}}^-(G) \subseteq {{\mathfrak {p}}}\). This shows that \({\mathfrak {p}}^-(G)\) and \({\mathfrak {p}}^+(G)\) are the minimal primes of \({\mathcal {J}}_{G}\).

If \(\mathrm{char}\,(\Bbbk ) = 2\), then [5, Corollary 2.2] gives that \({\mathcal {J}}_{G}\) is primary of multiplicity two over a minimal prime which equals \({\mathfrak {p}}^+(G) = {\mathfrak {p}}^-(G)\) by the above computation. It is now evident that the algebraic closure assumption on \(\Bbbk \) is irrelevant since all saturations of characters are defined over \(\Bbbk \). \(\square \)

Remark 4.3

The graph G is bipartite if and only if \({\mathcal {J}}_{G}\) is prime.

When decomposing a pure difference binomial ideal, all components except those over the saturation \({\mathcal {J}}_{G}\) contain monomials (for a combinatorial reason see [13, Example 4.14]). Our next step is to determine the indeterminates in the minimal primes. To this end, for any \(S\subseteq V(G)\) let \(G_S\) be the induced subgraph of G on \(V(G)\setminus S\) and \({\mathfrak {m}}_S:=\left\langle x_s,y_s: s\in S \right\rangle \).

Lemma 4.4

Let \({{\mathfrak {p}}}\) be a minimal prime of \({\mathcal {I}}_{G}\). Then, there exists \(S\subseteq V(G)\) and a minimal prime \({{\mathfrak {p}}}'\) of \({\mathcal {J}}_{G_S}\) such that \({{\mathfrak {p}}}={\mathfrak {m}}_S+{{\mathfrak {p}}}'\).

Proof

Let \(S:=\{s\in V(G): x_s,y_s\in {{\mathfrak {p}}}\}\). We first show the inclusions
$$\begin{aligned} {\mathcal {I}}_{G}\subseteq {\mathfrak {m}}_S + {\mathcal {J}}_{G_S} \subseteq {{\mathfrak {p}}}. \end{aligned}$$
The first inclusion is clear, while for the second, it suffices to check that \({\mathcal {J}}_{G_S} \subseteq {{\mathfrak {p}}}\). Generators of \({\mathcal {J}}_{G_S}\) correspond to (ij)-walks in \(G_S\) according to Proposition 2.7. Let b be the binomial corresponding to any such walk, and let \(\{k_1,\dots ,k_r\} \subseteq V(G)\setminus S\) be its interior. By Lemma 2.4, \(t_{k_1}\cdots t_{k_r}\cdot b\in {\mathcal {I}}_{G}\subseteq {{\mathfrak {p}}}\) for any choice of indeterminates \(t_{k_l} \in \{x_{k_l},y_{k_l}\}\), with  \(1\le l \le r\). By the construction of S, there exists some choice such that \(t_{k_1}\cdots t_{k_r} \notin {{\mathfrak {p}}}\). Since \({{\mathfrak {p}}}\) is prime, \(b\in {{\mathfrak {p}}}\). The minimal primes of \({\mathfrak {m}}_S + {\mathcal {J}}_{G_S}\) arise as sums of \({\mathfrak {m}}_S\) and minimal primes of \({\mathcal {J}}_{G_S}\). By minimality, \({{\mathfrak {p}}}\) equals \({\mathfrak {m}}_S+{{\mathfrak {p}}}'\) for some minimal prime \({{\mathfrak {p}}}'\) of \({\mathcal {J}}_{G}\). \(\square \)

Not all primes of the form \({\mathfrak {m}}_S+{{\mathfrak {p}}}'\) are minimal over \({\mathcal {I}}_{G}\) (see Example 4.10). As for binomial edge ideals, cut points play a crucial role in determining the sets S which lead to minimal primes, but for parity binomial edge ideals we count connected components differently. The bipartite ones count double.

Definition 4.5

Let \({\mathfrak {s}}(G)=c_0(G)+c(G) = 2c_0(G) + c_1(G)\). A set \(S\subseteq V(G)\) is a disconnector of G if \({\mathfrak {s}}(G_S)>{\mathfrak {s}}(G_{S\setminus {\{s\}}})\) for every \(s\in S\).

Remark 4.6

The empty set is a disconnector of any graph, and disconnectors cannot contain isolated vertices.

Remark 4.7

If a graph G has no isolated vertices, then \({\mathfrak {s}}(G_{\{s\}})\ge {\mathfrak {s}}(G)\) for all \(s\,{\in }\,V(G)\) and according to Definition 4.5, a vertex s is a disconnector of G exactly if the inequality is strict. Moreover, one can conclude from the following proposition that s is a disconnector of G if and only if \({\mathcal {J}}_{G}\not \subseteq {\mathfrak {m}}_{\{s\}}+{\mathcal {J}}_{G_{\{s\}}}\).

Proposition 4.8

Let G be a graph and \(S\subseteq V(G)\). Then, \({\mathcal {J}}_{G}\subseteq {\mathfrak {m}}_S+{\mathcal {J}}_{G_S}\) if and only if for all (ij)-walks in G with \(i,j\in V(G_S)\), there is an (ij)-walk in \(G_S\) of the same parity.

Proof

Let \({\mathcal {J}}_{G}\subseteq {\mathfrak {m}}_S+{\mathcal {J}}_{G_S}\). Let \(m\in {\mathcal {J}}_{G}\) be a Graver move corresponding to an (ij)-walk in G with \(i,j\not \in S\). Since \(m \in \Bbbk [x_i,x_j,y_i,y_j]\), and no polynomial in \({\mathcal {J}}_{G_S}\) uses indeterminates from S, we find \(m \in {\mathcal {J}}_{G_S}\). It follows that m is an element of the Graver basis of \({\mathcal {J}}_{G_S}\) and thus corresponds to an (ij)-walk in \(G_S\) of the same parity.

On the other hand, let \(m\in {\mathcal {J}}_{G}\) be a move corresponding to a (ij)-walk in G. If \(i\in S\) or \(j\in S\), then \(m\in {\mathfrak {m}}_S\). If otherwise \(i,j\in V(G_S)\), then \(m\in {\mathcal {J}}_{G_S}\) by assumption. \(\square \)

The next lemma states that the indeterminates contained in a minimal prime correspond to a disconnector of G, and Theorem 4.15 below says when the converse is true as well.

Lemma 4.9

Let \({{\mathfrak {p}}}\) be a minimal prime of \({\mathcal {I}}_{G}\). There exists a disconnector \(S\subseteq V(G)\) of G and a minimal prime \({{\mathfrak {p}}}'\) of \({\mathcal {J}}_{G_S}\) such that \({{\mathfrak {p}}}={\mathfrak {m}}_S+{{\mathfrak {p}}}'\).

Proof

Let S and \({{\mathfrak {p}}}'\) be as in Lemma 4.4. We prove that S is a disconnector. Assume the converse, i.e., there exists \(s\in S\) such that \(\{s\}\) is not a disconnector of \(G_{S\setminus \{s\}}\). According to Remark 4.7 and Proposition 4.8,
$$\begin{aligned} {\mathcal {J}}_{G_{S\setminus \{s\}}}\subseteq {\mathfrak {m}}_{\{s\}}+{\mathcal {J}}_{G_S}\subseteq {\mathfrak {m}}_{\{s\}}+{{\mathfrak {p}}}'. \end{aligned}$$
Hence, since the ideal on the right-hand side is prime, choose a minimal prime \({{\mathfrak {p}}}''\) of \({\mathcal {J}}_{G_{S\setminus \{s\}}}\) such that \({\mathcal {J}}_{G_{S\setminus \{s\}}}\subseteq {{\mathfrak {p}}}''\subsetneq {\mathfrak {m}}_{\{s\}}+{{\mathfrak {p}}}'\). This give rise to
$$\begin{aligned} {\mathcal {I}}_{G}\subseteq {\mathfrak {m}}_{S\setminus \{s\}}+{{\mathfrak {p}}}''\subsetneq {\mathfrak {m}}_{S}+{{\mathfrak {p}}}' = {{\mathfrak {p}}}\end{aligned}$$
which contradicts the minimality of \({{\mathfrak {p}}}\). \(\square \)
Let \(S\subseteq V(G)\) be a disconnector of G. The induced subgraph \(G_S\) splits into bipartite components \(B_1,\dots ,B_{c_0(G_S)}\) and non-bipartite components \(N_1,\dots ,N_{c_1(G_S)}\). By Proposition 4.2, the minimal primes of \({\mathcal {J}}_{G_S}\) are
$$\begin{aligned} {{\mathfrak {p}}}=\sum _{i=1}^{c_0(G_S)}{\mathcal {J}}_{B_i} + \sum _{i=1}^{c_1(G_S)}{{\mathfrak {p}}}^{\sigma _i}(N_i), \text { where } {\left\{ \begin{array}{ll} \sigma _i \in \{+,-\}, &{} \text { if }\mathrm{char}\,(\Bbbk )\ne 2,\\ \sigma _i = +, &{} \text { if }\mathrm{char}\,(\Bbbk )= 2. \end{array}\right. } \end{aligned}$$
(4.1)
Not all of these primes lead to minimal primes of \({\mathcal {I}}_{G}\) because of the following effect.

Example 4.10

Let G be the graph in Fig. 2. The vertex 4 is a disconnector, and \(G_{\{4\}}\) consists of the two triangles \(N_1=\{1,2,3\}\) and \(N_2=\{5,6,7\}\). Choosing for both triangles the positive sign component, we obtain the prime ideal
$$\begin{aligned} {\mathfrak {m}}_{\{4\}}+{\mathfrak {p}}^+(N_1)+{\mathfrak {p}}^+(N_2)={\mathfrak {m}}_{\{4\}}+\left\langle x_i+y_i: i\in [7]\setminus \{4\} \right\rangle \end{aligned}$$
which is not minimal over \({\mathcal {I}}_{G}\) since it contains the prime ideal \({\mathfrak {p}}^+(G)\). On the other hand, both ideals with the binomial part \({\mathfrak {m}}_{\{4\}}+{{\mathfrak {p}}}^\pm (N_1)+{{\mathfrak {p}}}^\mp (N_2)\), each having different signs on the triangles, are minimal over \({\mathcal {I}}_{G}\).
Fig. 2

A graph for which one of the primes in (4.1) is not a minimal prime

A combinatorial condition on \(\sigma \) in (4.1) guarantees that a minimal prime of \({\mathcal {J}}_{G_S}\) is the binomial part of a minimal prime of \({\mathcal {I}}_{G}\) (the monomial part being \({\mathfrak {m}}_S\)). To see it, let \(s\in S\) be such that \(c(G_S)>c(G_{S\setminus \{s\}})\), i.e., when adding s back to \(G_S\) some of its connected components are joined. Denote by \({{\mathcal {C}}}_{G_S}(s)\) the set of only those connected components of \(G_S\) which are joined when adding s.

Definition 4.11

Let \(S\subseteq V(G)\) be a disconnector of G. A minimal prime \({{\mathfrak {p}}}\) of \({\mathcal {J}}_{G_S}\) is sign-split if for all \(s\in S\) such that \({{\mathcal {C}}}_{G_S}(s)\) contains no bipartite graphs, the prime summands of \({{\mathfrak {p}}}\) corresponding to connected components in \({{\mathcal {C}}}_{G_S}(s)\) are not all equal to \({{\mathfrak {p}}}^+\) or all equal to \({{\mathfrak {p}}}^-\).

Remark 4.12

If \({{\mathcal {C}}}_{G_S}(s)\) contains at least one bipartite graph, then Definition 4.11 imposes no restriction and every choice of prime summands is sign-split.

Remark 4.13

If \(\mathrm{char}\,(\Bbbk ) = 2\), then all signs \(\sigma \) in (4.1) are fixed. In this case, Definition 4.11 can only be satisfied if \({{\mathcal {C}}}_{G_S}(s)\) contains a bipartite component for each \(s\in S\).

Example 4.14

Not every disconnector \(S\subseteq V(G)\) of G admits a sign-split minimal prime for \({\mathcal {J}}_{G_S}\), and thus not every disconnector contributes minimal primes to \({\mathcal {I}}_{G}\). Consider the graph in Fig. 3. The set of blue square vertices is a disconnector that does not contribute minimal primes. Adding one of the squares back yields the requirement that the primes on the two now connected triangles have different signs, but these three requirements cannot be satisfied simultaneously.

Fig. 3

A disconnector whose binomial parts cannot be sign-split

Theorem 4.15

The minimal primes of \({\mathcal {I}}_{G}\) are the ideals \({\mathfrak {m}}_S+{{\mathfrak {p}}}\), where \(S\subseteq V(G)\) is a disconnector of G and \({{\mathfrak {p}}}\) is a sign-split minimal prime of \({\mathcal {J}}_{G_S}\).

Proof

According to Lemma 4.9, all minimal primes of \({\mathcal {I}}_{G}\) have the form \({\mathfrak {m}}_S+{{\mathfrak {p}}}\), where \(S\subseteq V(G)\) is a disconnector and \({{\mathfrak {p}}}\) is a minimal prime of \({\mathcal {J}}_{G_S}\). We first show that if \({{\mathfrak {p}}}\) is sign-split, this ideal is minimal over \({\mathcal {I}}_{G}\). Assume not, then by Lemma 4.4 there exists a set \(T\subseteq V(G)\) and a minimal prime \({\tilde{{{\mathfrak {p}}}}}\) of \({\mathcal {J}}_{G_T}\) such that
$$\begin{aligned} {\mathcal {I}}_{G}\subseteq {\mathfrak {m}}_T+{\tilde{{{\mathfrak {p}}}}}\subsetneq {\mathfrak {m}}_S+{{\mathfrak {p}}}. \end{aligned}$$
(4.2)
This implies \(T\subsetneq S\), since if \(T=S\), then by Lemma 4.4 also \({\tilde{{{\mathfrak {p}}}}}={{\mathfrak {p}}}\). Let \(s'\in S\setminus T\), then \(G_S\subsetneq G_{S\setminus \{s'\}}\subseteq G_T\). Since \(s'\) is a disconnector of \(G_{S\setminus \{s'\}}\), \({\mathfrak {s}}(G_{S})>{\mathfrak {s}}(G_{S\setminus \{s'\}})\). Let again \({{\mathcal {C}}}_{G_S}(s')\) be the set of connected components in \(G_S\) that are joined to \(s'\) in \(G_{S\setminus \{s'\}}\). If \({{\mathcal {C}}}_{G_S}(s')\) contains at least one bipartite component, adding \(s'\) to \(G_S\) either this component becomes non-bipartite in \(G_{S\setminus \{s'\}}\) or it is joined to another bipartite component of \(G_S\). In the first case, let B be a bipartite component which becomes non-bipartite. There exists \(i\in V(B)\) such that \(x_i^2-y_i^2\in {\mathcal {J}}_{G_{S\setminus \{s'\}}}\subseteq {\mathcal {J}}_{G_T}\subseteq {\tilde{{{\mathfrak {p}}}}}\), but \(x_i^2-y_i^2\not \in {\mathcal {J}}_{B}\). Since \({\mathcal {J}}_{B}\) is a summand of \({{\mathfrak {p}}}\), \(x_i^2-y_i^2\not \in {\mathfrak {m}}_S+{{\mathfrak {p}}}\), in contradiction to (4.2). In the second case, let \(B_1\) and \(B_2\) be the bipartite components of \(G_S\) which are joined to \(s'\). There are \(i_1\in V(B_1)\) and \(i_2\in V(B_2)\) such that there exists an \((i_1,i_2)\)-walk in \(G_{S\setminus \{s'\}}\). Independent of the parity of this walk, the corresponding Markov move is not contained in \({\mathcal {J}}_{B_1}+{\mathcal {J}}_{B_2}\) since there is no applicable move from the Graver basis. Since \({\mathcal {J}}_{B_1}\) and \({\mathcal {J}}_{B_2}\) are summands of \({{\mathfrak {p}}}\) involving the indeterminates \(i_1\) and \(i_2\), there is a binomial which is not in \({\mathfrak {m}}_S+{{\mathfrak {p}}}\) but in \({\mathcal {J}}_{G_{S\setminus \{s'\}}}\subseteq {\tilde{{{\mathfrak {p}}}}}\) contradicting (4.2).
Assume now that all components in \({{\mathcal {C}}}_{G_S}(s')\) are non-bipartite (there are at least two of them since \(s'\) is a disconnector). By assumption, \({{\mathfrak {p}}}\) is sign-split, i.e., there exist distinct components \(N_1,N_2\in {{\mathcal {C}}}_{G_S}(s)\) such that \({\mathfrak {p}}^+(N_1)\) and \({\mathfrak {p}}^-(N_2)\) are summands of \({{\mathfrak {p}}}\). There is an odd walk from a vertex \(i_1\in V(N_1)\) to a vertex \(i_2\in V(N_2)\) in \(G_{S\setminus \{s'\}}\), and therefore, \(x_{i_1}x_{i_2}-y_{i_1}y_{i_2}\in {\mathcal {J}}_{G_{S\setminus \{s'\}}}\subseteq {\tilde{{{\mathfrak {p}}}}}\). Since
$$\begin{aligned} x_{i_1}x_{i_2}-y_{i_1}y_{i_2}\not \in {\mathfrak {p}}^+(N_1)+{\mathfrak {p}}^-(N_2), \end{aligned}$$
also \(x_{i_1}x_{i_2}-y_{i_1}y_{i_2}\not \in {{\mathfrak {p}}}\). By construction, \(i_1,i_2\not \in S\) and thus
$$\begin{aligned} x_{i_1}x_{i_2}-y_{i_1}y_{i_2}\not \in {\mathfrak {m}}_S+{{\mathfrak {p}}}\end{aligned}$$
which contradicts (4.2). This shows minimality of \({\mathfrak {m}}_S+{{\mathfrak {p}}}\).
Let now \({\mathfrak {m}}_S+{{\mathfrak {p}}}\) be a minimal prime of \({\mathcal {I}}_{G}\). The set S is a disconnector by Lemma 4.9, and thus, it remains to prove that \({{\mathfrak {p}}}\) is sign-split. To the contrary, assume there is a vertex \(s\in S\) with \(c(G_{S\setminus \{s\}})>c(G_S)\) such that \({{\mathcal {C}}}_{G_S}(s) = \{N_1,\ldots ,N_k\}\) consists exclusively of non-bipartite components, \(k \ge 2\), and all summands of \({{\mathfrak {p}}}\) corresponding to \(N_i\) have the same sign, say \(+\). When adding s back to \(G_{S}\), the components in \({{\mathcal {C}}}_{G_S}(s)\) are joined to a single, non-bipartite connected component H in \(G_{S\setminus \{s\}}\), whereas all other components of \(G_S\) coincide with connected components of \(G_{S\setminus \{s\}}\). Since
$$\begin{aligned} {\mathfrak {p}}^+(H)={\mathfrak {m}}_{\{s\}}+\sum _{i=1}^k{\mathfrak {p}}^+(N_i)\subsetneq {\mathfrak {m}}_{\{s\}}+\sum _{i=1}^k{\mathfrak {p}}^+(N_i), \end{aligned}$$
choosing on all other components of \(G_{S\setminus \{s\}}\) the same prime component as in \(G_S\), we obtain a prime ideal that is strictly smaller than \({\mathfrak {m}}_S+{{\mathfrak {p}}}\). \(\square \)

Remark 4.16

Example 4.10 and Definition 4.11 are valid independent of \(\mathrm{char}\,(\Bbbk )\). In the above proof, the case of \(\mathrm{char}\,(\Bbbk ) = 2\) could be simplified, but everything works in general without the need for a case distinction.

5 Radicality and mesoprimary decomposition

The intersection of the minimal primes of \({\mathcal {I}}_{G}\) depends on \(\mathrm{char}\,(\Bbbk )\) so that we do not attempt to compute it directly. Theorem 5.5 below says that \({\mathcal {I}}_{G}\) is radical if the characteristic is not two. Here is the principal source of field dependence (see also [8, Theorem 1.2]).

Remark 5.1

Fix a field \(\Bbbk \) with \(\mathrm{char}\,(\Bbbk )=2\). The parity binomial edge ideal \({\mathcal {I}}_{G}\) is radical in \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\) if and only if G is bipartite. Clearly, if G is bipartite, then \({\mathcal {I}}_{G}\) is radical by Remark 1.2. Conversely, let \((i_1,\ldots ,i_{r+1})\) with \(i_{r+1}=i_1\) be an odd cycle in G. According to Lemma 2.4, \(((x_{i_1}-y_{i_1})y_{i_2}\cdots y_{i_{r}})^2{=}(x_{i_1}^2-y_{i_1}^2)y_{i_2}^2\cdots y_{i_{r}}^2\,{\in }\,{\mathcal {I}}_{G}\). The only possible reduced binomials whose leading monomials divide \(x_{i_1}y_{i_2}\cdots y_{i_{r}}\) correspond to minimal even \((i_1,i_k)\)-walks in \(G[\{i_1,\ldots ,i_{r}\}]\) with \(k\in \{2,\ldots ,r\}\). Replacements coming from these binomials lead to monomials where \(x_{i_1}\) is replaced by \(y_{i_1}\) and \(y_{i_k}\) is replaced by \(x_{i_k}\). Thus, \(x_{i_1}y_{i_2}\cdots y_{i_{r}}\not \equiv _{{\mathcal {I}}_{G}}y_{i_1}y_{i_2}\cdots y_{i_{r}}\) and hence \({\mathcal {I}}_{G}\) is not radical.

Remark 5.2

The ideal \({\mathcal {I}}_{G}\) is homogeneous with respect to the multigrading \(\deg (x_i)=\deg (y_i)=e_i\), where \(e_i\) is the ith standard basis vector of \(\mathbb {R}^{|V(G)|}\).

Lemma 5.3

Let \(i\in V(G)\) and \(m\in {\mathcal {I}}_{G} + {\mathfrak {m}}_{\{i\}}\) be a monomial. Then, \(m\in {\mathfrak {m}}_{\{i\}}\).

Proof

Since it is generated by pure difference binomials, \({\mathcal {I}}_{G}\) does not contain any monomials. Thus, any monomial in \({\mathcal {I}}_{G} + {\mathfrak {m}}_{\{i\}}\) is equivalent to one in \({\mathfrak {m}}_{\{i\}}\) modulo term replacements using binomials in \({\mathcal {I}}_{G}\), but these do not change membership in \({\mathfrak {m}}_{\{i\}}\) by Remark 5.2. \(\square \)

Proposition 5.4

For any graph G, \({\mathcal {I}}_{G}={\mathcal {J}}_{G}\cap \bigcap _{i\in V(G)}({\mathcal {I}}_{G}+{\mathfrak {m}}_{\{i\}}).\)

Proof

According to [5, Corollary 1.5], the intersection is binomial. Let b be any binomial in the intersection. For each \(i\in V(G)\), there are three cases: Either no term of b is individually contained in \({\mathcal {I}}_{G} + {\mathfrak {m}}_{\{i\}}\), exactly one is, or both are. In the first case, [5, Proposition 1.10] implies \(b \in {\mathcal {I}}_{G}\). In the second case, it implies that the other monomial is contained in \({\mathcal {I}}_{G}\), which is impossible. Thus, it suffices to consider binomials b both of whose monomials are contained in \({\mathcal {I}}_{G} + {\mathfrak {m}}_{\{i\}}\) for all \(i\in V(G)\). By Lemma 5.3, both monomials of b are contained in \({\mathfrak {m}}_{\{i\}}\) for each \(i\in V(G)\). Since \(b\,{\in }\,{\mathcal {J}}_{G}\), there exist Markov moves \(m_{s_1t_1},\ldots ,m_{s_rt_r}\) corresponding to \((s_1,t_1),\ldots ,(s_r,t_r)\)-walks, respectively, such that
$$\begin{aligned} b=x^{h_1}y^{h'_1}m_{s_1t_1}+\cdots +x^{h_r}y^{h'_r}m_{s_rt_r} \end{aligned}$$
with \(h_i,h_i'\in \mathbb {N}^n\). We can assume that one monomial of b equals one of the monomials of \(x^{h_1}y^{h_1}m_{s_1t_1}\). Thus, both monomials of \(x^{h_1}y^{h_1}m_{s_1t_1}\) are divisible by at least one indeterminate for each \(i\in V(G)\) and, by Lemma 2.4, \(x^{h_1}y^{h_1}m_{s_1t_1} \in {\mathcal {I}}_{G}\). Replacing b by \(b-x^{h_1}y^{h_1}m_{s_1t_1}\) and iterating the argument eventually yields \(b\in {\mathcal {I}}_{G}\). \(\square \)

Theorem 5.5

Let G be a graph. If \( char (\Bbbk )\ne 2\), then \({\mathcal {I}}_{G}\) is a radical ideal.

Proof

The proof is by induction on the number of vertices of G. If G has at most one vertex, then \({\mathcal {I}}_{G}=0\) and the claim holds. Remark 5.2 shows that \({\mathcal {I}}_{G}+{\mathfrak {m}}_{\{i\}}\,{=}\,{\mathcal {I}}_{G_{\{i\}}}\,{+}\,{\mathfrak {m}}_{\{i\}}\) for all \(i\in V(G)\). Thus, Theorem 5.4 reads as \({\mathcal {I}}_{G}={\mathcal {J}}_{G}\cap \bigcap _{i=1}^n({\mathcal {I}}_{G_{\{i\}}}+{\mathfrak {m}}_{\{i\}})\). By the induction hypothesis, \({\mathcal {I}}_{G_{\{i\}}}\) is radical and thus \({\mathcal {I}}_{G_{\{i\}}}+{\mathfrak {m}}_{\{i\}}\) is radical. Remark 4.3 says that \({\mathcal {J}}_{G}\) is radical if and only if \(\mathrm{char}\,(\Bbbk ) \ne 2\) which yields the result. \(\square \)

Theorem 5.9 below contains a primary decomposition of \({\mathcal {I}}_{G}\) in the case \(\mathrm{char}\,(\Bbbk ) = 2\). It uses the following lemma, which allows to transport decompositions between different characteristics. Recall that the combinatorics of any binomial ideal I is encoded in its congruence \(\mathord \sim _I\) which identifies monomials \(m_1,m_2\), whenever \(m_1-\lambda m_2 \in I\) for some nonzero \(\lambda \in \Bbbk \). A binomial ideal is unital if it is generated by monomials and pure differences of monomials. Then, each congruence is the congruence of a unital binomial ideal, though not uniquely.

Lemma 5.6

If a decomposition \(I = J_1 \cap \dots \cap J_s\) of a unital binomial ideal I into unital binomial ideals \(J_i\), \(i=1,\dots ,s\) is valid in some characteristic, then it is valid in any characteristic.

Proof

The congruence \(\mathord \sim _I\) induced by I is the common refinement of the congruences \(\mathord \sim _{J_i}\), induced by the \(J_i\), \(i=1,\dots , s\). Thus, in any characteristic, [13, Theorem 9.12] implies that I and \(J_1\cap \dots \cap J_s\) can only differ if one of them contains monomials, but the other does not. This cannot happen since unital binomial ideals contain monomials if and only if they have monomials among the generators. \(\square \)

According to Example 4.14, not all disconnectors contribute minimal primes. From Definition 4.11, it may seem that this is an arithmetic effect. It is not; the primary decomposition of \({\mathcal {I}}_{G}\) in characteristic two also witnesses it. For the following definition, recall that a hypergraph is (k-colorable) if the vertices can be colored with k colors so that no edge is monochromatic.

Definition 5.7

Let \(S \subseteq V(G)\) be a disconnector, and let \(s_1,\ldots ,s_r\in S\) be the vertices such that \({{\mathcal {C}}}_{G_S}(s_i)\) consists exclusively of non-bipartite components of \(G_S\). Let \({\mathcal {H}}\) be the hypergraph whose vertex set consists of the connected components \({{\mathcal {C}}}_{G_S}(s_1)\cup \ldots \cup {{\mathcal {C}}}_{G_S}(s_r)\) and with edge set \(\{{{\mathcal {C}}}_{G_S}(s_1),\ldots ,{{\mathcal {C}}}_{G_S}(s_r)\}\). The disconnector S is effective if \({\mathcal {H}}\) is 2-colorable.

Remark 5.8

A disconnector is effective if and only if, in characteristic zero, it admits sign-split minimal primes.

Theorem 5.9

Let \({\mathcal {S}}\) be the set of effective disconnectors of G. Then,
$$\begin{aligned} {\mathcal {I}}_{G} = \bigcap _{S\in {\mathcal {S}}}\left( {\mathfrak {m}}_S + {\mathcal {J}}_{G_S}\right) . \end{aligned}$$
(5.1)
If \(\mathrm{char}\,(\Bbbk )=2\), then (5.1) is a primary decomposition of \({\mathcal {I}}_{G}\).

Proof

For each disconnector \(S\in {\mathcal {S}}\), let \(B^S_1,\dots ,B^S_{c_0(G_S)}\) be the bipartite components and \(N^S_1,\dots ,N^S_{c_1(G_S)}\) the non-bipartite components of \(G_S\). Let \(\Sigma ^S \subseteq \{+,-\}^{c_1(G_S)}\) denote the set of sign patterns that are sign-split. In characteristic zero, by Theorems 4.15 and 5.5, \({\mathcal {I}}_{G}\) decomposes as
$$\begin{aligned} {\mathcal {I}}_{G} = \bigcap _{S\in {\mathcal {S}}} \bigcap _{\sigma \in \Sigma ^S} \left( {\mathfrak {m}}_S + \sum _{i=1}^{c_0(G_S)} {\mathcal {J}}_{B^S_i} + \sum _{i=1}^{c_1(G_S)} {{\mathfrak {p}}}^{\sigma _i}(N^S_i) \right) . \end{aligned}$$
The intersection remains valid when intersecting over additional ideals containing \({\mathcal {I}}_{G}\). In particular, the sign-split requirement can be dropped and \(\Sigma ^S\) replaced by \(\{+,-\}^{c_1(G_S)}\). Carrying out this inner intersection yields the ideals \({\mathfrak {m}}_S+{\mathcal {J}}_{G_S}\) by Proposition 4.2, and hence, (5.1) is valid in characteristic zero. Since all involved ideals are unital, Lemma 5.6 yields that (5.1) is valid in any characteristic. The ideals under consideration are primary if \(\mathrm{char}\,(\Bbbk )=2\) according to Proposition 4.2, and thus, the second statement follows. \(\square \)

The technique of adding “phantom components” to a primary decomposition so that it faithfully exists over some other field (as we did in the proof of Theorem 5.9) was mentioned in the introduction of [13] as one way to arrive at more combinatorially accurate decompositions of binomial ideals. The upshot of the rest of the paper is that the primary decomposition in characteristic two is an example of a mesoprimary decomposition. This means not only that all ideals in (5.1) are mesoprimary, but additionally each of the intersectands witnesses a combinatorial feature of the graph that the binomials of \({\mathcal {I}}_{G}\) induce on the monomials of \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\). Generally, it can be quite challenging to determine a mesoprimary decomposition, exactly because of the stringent combinatorial conditions that it has to meet. Here, it is mostly a translation of the (involved) definitions, essentially because all ideals are unital and the ambient ring is the polynomial ring [13, Remark 12.8]. We refrain from introducing too much of the machinery from [13] here, but do employ their notation. In the following, we give explicit references to all relevant definitions.

Theorem 5.10

The decomposition (5.1) is a mesoprimary decomposition of \({\mathcal {I}}_{G}\).

Proof

For \(S\in {\mathcal {S}}\), let \(J_S:={\mathfrak {m}}_S+{\mathcal {J}}_{G_S}\) be the intersectand corresponding to S. The ideal \(J_S\) is \(P_S\)-mesoprimary where \(P_S \subseteq \mathbb {N}^{2|V(G)|}\) is the monoid prime ideal \(\langle e_{x_i},e_{y_i} : i \in S\rangle \). In fact (like any ideal that equals a lattice ideal plus monomials in a disjoint set of indeterminates), \(J_S\) is mesoprime [13, Definition 10.4] since it equals the kernel of the monomial homomorphism
$$\begin{aligned} \Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}] \rightarrow \Bbbk [x_i^\pm ,y_i^\pm : i\notin S]/{\mathcal {J}}_{G_S} = \Bbbk [\mathbb {Z}^{2|V(G) \setminus S|}/L_s] \end{aligned}$$
which maps \(x_i,y_i\) to zero if \(i\in S\) and to their images in the Laurent ring if \(i\notin S\). Here \(L_S\) is the image in \(\mathbb {Z}^{2|V(G) \setminus S|}\) of the adjacency matrix of \(G_S\).

According to [13, Definition 13.1], it remains to show that at each cogenerator [13, Definitions 7.1 and 12.16] of \(J_S\), the \(P_S\)-mesoprimes of \({\mathcal {I}}_{G}\) and \(J_S\) agree. Let \(J_S^\pm \) be the image of \(J_S\) in \(R^S = \Bbbk [x_i,y_i, i\in S, x^\pm _j,y^\pm _j, j\notin S]\). The cogenerators of \(J_S\) are monomials in \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\) whose images in \(R_S/J_S^\pm \) are annihilated by \({\mathfrak {m}}_S\). Since \(J_S\) contains \({\mathfrak {m}}_S\), the cogenerators are simply all monomials in the indeterminates \(x_i,y_i\) for \(i\notin S\). Now, the \(P_S\)-mesoprime of the mesoprime \(J_S\) (at any monomial) is just \(J_S\). Thus, it remains to compute the \(P_S\)-mesoprime of \({\mathcal {I}}_{G}\) at any cogenerator. Translating [13, Definition 11.11] to \(\Bbbk [{{\mathbf {x}}},{{\mathbf {y}}}]\), this mesoprime is given by \(\left( {\mathcal {I}}_{G} + {\mathfrak {m}}_S\right) : \left( \prod _{i\notin S} x_iy_i\right) ^\infty \). The result now follows by Lemma 5.3. \(\square \)

The stringent combinatorial conditions that guarantee a canonical mesoprimary decomposition require additional knowledge about the witness structure of \({\mathcal {I}}_{G}\).

Conjecture 5.11

The mesoprimary decomposition in (5.1) is combinatorial and characteristic.

To prove Conjecture 5.11, one needs precise control over the various witnesses that contribute to coprincipal decompositions [13, Theorems 8.4 and 16.9]. Experiments with Macaulay2 indicate that the mesoprimary decomposition of the congruence \(\mathord \sim _{{\mathcal {I}}_{G}}\) differs significantly from that of the ideal \({\mathcal {I}}_{G}\). For example, if G is a path \(G = 1-2-3-4-5\), then \({\mathcal {I}}_{G}\) has the following mesoprimary decomposition:
$$\begin{aligned} {\mathcal {I}}_{G}&= {\mathcal {J}}_{G} \cap ({\mathfrak {m}}_{\{4\}} + {\mathcal {J}}_{1-2-3}) \cap ({\mathfrak {m}}_{\{2\}} + {\mathcal {J}}_{3-4-5}) \\&\ \cap ({\mathfrak {m}}_{\{3\}} + {\mathcal {I}}_{1-2} + {\mathcal {I}}_{4-5}) \cap {\mathfrak {m}}_{\{2,4\}}. \end{aligned}$$
Intersecting all but the last ideal yields the ideal
$$\begin{aligned} {\mathcal {I}}_{G} + \left\langle x_1x_3y_3y_5 - x_1x_5y_3^2 - x_3^2 y_1y_5 + x_3x_5y_1y_3 \right\rangle . \end{aligned}$$
This ideal has the same binomials as \({\mathcal {I}}_{G}\) and thus induces the same congruence. The monomial ideal that was omitted does not influence the congruence. Its sole purpose is to cut away non-binomials. The monoid prime \(\langle e_{x_2},e_{y_2},e_{x_4},e_{y_4}\rangle \subseteq \mathbb {N}^{2|V(G)|}\) contributes only non-key witnesses to the case. Nevertheless, these witnesses are essential in the sense of [13, Definition 12.1].

Notes

Acknowledgments

The authors would like to thank Rafael Villareal for posting the question of radicality of parity binomial edge ideals. We thank Fatemeh Mohammadi for pointing us at [9]. The authors appreciate the many comments and suggestions by Issac Burke and Mourtadha Badiane. T.K. and C.S. are supported by the Center for Dynamical Systems (CDS) at Otto-von-Guericke University Magdeburg. T.W. is supported by the German National Academic Foundation and TopMath, a graduate program of the Elite Network of Bavaria.

References

  1. 1.
    Bayer, D., Popescu, S., Sturmfels, B.: Syzygies of unimodular Lawrence ideals. Journal für die Reine und Angewandte Mathematik 534, 169–186 (2001)MathSciNetMATHGoogle Scholar
  2. 2.
    Conradi, C., Kahle, T.: Detecting binomiality. Adv. Appl. Math. 71, 52–67 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and Geometric Ideas in the Theory of Discrete Optimization. MPS-SIAM Series on Optimization, SIAM, Cambridge (2013)MATHGoogle Scholar
  4. 4.
    Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics, Oberwolfach Seminars, A Birkhäuser book, vol. 39, Springer, Berlin (2009)Google Scholar
  5. 5.
    Eisenbud, D., Sturmfels, B.: Binomial ideals. Duke Math. J. 84(1), 1–45 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  7. 7.
    Grossman, J.W., Kulkarni, D.M., Schochetman, I.E.: On the minors of an incidence matrix and its Smith normal form. Linear Algebra Appl. 218(1995), 213–224 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Herzog, J., Hibi, T., Hreinsdóttir, F., Kahle, T., Rauh, J.: Binomial edge ideals and conditional independence statements. Adv. Appl. Math. 45(3), 317–333 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Herzog, J., Macchia, A., Madani, S.S., Welker, V.: On the ideal of orthogonal representations of a graph. Adv. Appl. Math. 71, 146–173 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hoşten, S., Shapiro, J.: Primary decomposition of lattice basis ideals. J. Symb. Comput. 29(4–5), 625–639 (2000)MathSciNetMATHGoogle Scholar
  11. 11.
    Hoşten, S., Sullivant, S.: Ideals of adjacent minors. J. Algebra 277(2), 615–642 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kahle, T.: Decompositions of binomial ideals. J. Softw. Algebra Geom. 4, 1–5 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kahle, T., Miller, E.: Decompositions of commutative monoid congruences and binomial ideals. Algebra Number Theory 8(6), 1297–1364 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kahle, T., Rauh, J., Sullivant, S.: Positive margins and primary decomposition. J. Commut. Algebra 6(2), 173–208 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Miller, E.: Theory and applications of lattice point methods for binomial ideals. In: Proceedings of the Abel Symposium held at Voss, Norway, 2009, pp. 99–154. Springer (2011)Google Scholar
  16. 16.
    Miller, E.: Affine stratifications from finite misère quotients. J. Algebraic Comb. 37(1), 1–9 (2013)CrossRefMATHGoogle Scholar
  17. 17.
    Pérez Millán, M., Dickenstein, A., Shiu, A., Conradi, C.: Chemical reaction systems with toric steady states. Bull. Math. Biol. 74, 1027–1065 (2012)Google Scholar
  18. 18.
    Rauh, J., Sullivant, S.: Lifting Markov bases and higher codimension toric fiber products. J. Symb. Comput. 74, 276–307 (2016). doi:10.1016/j.jsc.2015.07.003 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Swanson, I.: On the embedded primes of the Mayr–Meyer ideals. J. Algebra 275, 143–190 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Windisch, T.: BinomialEdgeIdeals, a Macaulay2 package for (parity) binomial edge ideals.https://github.com/windisch/BinomialEdgeIdeals

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Thomas Kahle
    • 1
  • Camilo Sarmiento
    • 1
  • Tobias Windisch
    • 1
  1. 1.Otto-von-Guericke Universität MagdeburgMagdeburgGermany

Personalised recommendations