Multigraded commutative algebra of graph decompositions
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Abstract
The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in nonzero codimension and discuss persistence of normality and primary decompositions under toric fiber products.
Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K _{4}minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.
Keywords
Toric fiber product Toric ideal Segre product Markov basis Primary decomposition Algebraic statistics Conditional independence ideal Normality of ideals1 Introduction
Let I and J be ideals in polynomial rings \(\mathbb{K}[x]\) and \(\mathbb{K}[y]\), respectively, that are both homogeneous with respect to a single grading by an affine semigroup \(\mathbb{N}\mathcal{A}\). The toric fiber product of I and J (Definition 2.1), denoted \(I \times_{\mathcal{A}}J\), is a new ideal in a usually larger polynomial ring \(\mathbb{K}[z]\). An important measure of complexity of this operation is the codimension of the product, defined as the rank of the integer lattice \(\ker\mathcal{A}\). In [34] the third author introduced toric fiber products and proved that in the codimension zero case it is possible to construct a generating set or Gröbner basis for \(I \times _{\mathcal{A}}J\) from generating sets or Gröbner bases of I and J. In this case the algebra and geometry is significantly simpler essentially because codimension zero toric fiber products are multigraded Segre products (Definition 2.3), which share many nice properties with their standard graded analogues. Still in the codimension zero case, the geometry of the toric fiber product can be understood quite explicitly in terms of GIT [25] (Propositions 2.2 and 2.4). We pursue this observation and show that (under mild assumptions on \(\mathbb{K}\)) normality persists (Theorem 2.5).
The main goal of this paper, however, is to describe higher codimension toric fiber products. In Sect. 3 we show that primary decompositions persist in any codimension (Theorem 3.1). In Sect. 4 we show how to construct generating sets of toric fiber products in arbitrary codimension, but under some extra technical conditions (Theorem 4.9). This generalizes the codimension one results on cut ideals obtained by the first author in [11].
The toric fiber product frequently appears in applications of combinatorial commutative algebra, in particular in algebraic statistics [12, 31, 32]. Typically in algebraic statistics, we are interested in studying a family of ideals, where each ideal I _{ G } is associated to a graph G (or other combinatorial object, like a simplicial complex or a poset). If the graph has a decomposition into two simpler graphs G _{1} and G _{2}, we would like to show that the ideal I _{ G } has a decomposition into the two ideals \(I_{G_{1}}\) and \(I_{G_{2}}\). If we can identify I _{ G } as a toric fiber product \(I_{G_{1}} \times_{\mathcal{A}}I_{G_{2}}\), then difficult algebraic questions for large graphs reduce to simpler problems on smaller graphs. Our inspiration comes from structural graph theory, where the imposition of forbidden substructures often implies that a graph has a specific kind of structural decomposition into simple pieces. In Sect. 5 we pursue the analogy to the theory of forbidden minors [29] by exhibiting minorclosed classes of graphs with certain degree bounds on their Markov bases.
Before proving our main theoretical results in Sects. 2–4, we motivate our study with several examples from algebraic statistics. Sections 5 and 6 contain new applications to the construction of Markov bases of hierarchical models, and to the study of primary decompositions of conditional independence ideals.
1.1 Hierarchical models
Hierarchical statistical models are used to analyze associations between collections of random variables. If the random variables are discrete, these models are toric varieties, and hence their vanishing ideals are toric ideals. Their binomial generators—known as Markov bases—are useful for performing various tests in statistics [6, 10]. From the algebraic standpoint, they are binomial ideals with a specific combinatorial parametrization in terms of a simplicial complex.
Theorem 1.1
Let d _{ i }=2 for all i∈V and let Γ be a graph with no K _{4} minors. Then I _{ Γ,d } is generated by binomials of degrees two and four.
Combining our techniques with results from [15], we can also make statements about the asymptotic behavior as the d _{ i } grow. For instance, let F⊆V be an independent set of Γ and consider I _{ Γ,d } as d _{ i } tend to infinity for i∈F, while the remaining d _{ i } are fixed. In this case, there is a bound M(Γ,d _{ V∖F }) for the degrees of elements in minimal generating sets of I _{ Γ,d }. Our techniques allow us to determine the values of M(Γ,d _{ V∖F }), which were previously known only for reducible models or when F is a singleton [17]. Here is a simple example of how to apply Theorem 5.15.
Example 1.2
1.2 Conditional independence
If G is a graph on V, then its clique complex defines a hierarchical model as in the previous section. Probability distributions in this hierarchical model satisfy certain conditional independence statements associated to the graph [22]. One may ask which other distributions outside the hierarchical model also satisfy the conditional independence constraints, and algebraic statistics allows one to characterize these distributions. Consider again the polynomial ring \(\mathbb{K}[p_{i} : i \in\mathrm{D}_{V}]\) with one indeterminate for each elementary probability. If A,B,C⊂V is a partition of V, i.e. pairwise disjoint with A∪B∪C=V, the conditional independence (CI)statement Open image in new window encodes that the random variables in A are independent of the random variables in B, given the values of the random variables in C. Distributions satisfying this constraint form a hierarchical model, which arises from the largest simplicial complex on V not containing \(\left\lbrace i,j \right\rbrace\) for any i∈A,j∈B. Its toric ideal is denoted Open image in new window . A conditional independence model usually contains several statements and one is led to consider intersections of toric varieties. Our main interest is in the global Markov ideal of a graph G, which is the sum of the toric ideals Open image in new window for all A,B,C forming a partition of V such that C separates A and B in G. Our goal is to determine primary decompositions and as always we want to employ the toric fiber product machinery to split the problem into several easier problems.
Example 1.3
A systematic check of all graphs with at most five vertices and with d _{ v }=2 for all v∈V found no examples of a nonradical global Markov ideal. This limited computational evidence motivates the following question:
Question 1.4
Are global Markov ideals always radical?
The answer to this question is negative. More than a year after first submission of the present paper, Kahle, Rauh, and Sullivant showed that the global Markov ideal of K _{3,3} is not radical [20].
2 Toric fiber products and multigraded Segre products
Definition 2.1
We can also define the \(\mathbb{K}\)algebra homomorphism \(\phi: \mathbb{K}[z] \rightarrow \mathbb{K}[x] \otimes_{\mathbb{K}}\mathbb{K}[y] = \mathbb{K}[x,y]\) by \(z^{i}_{jk} \mapsto x^{i}_{j} y^{i}_{k}\). Then the toric fiber product is the ideal \(I\times_{\mathcal{A}}J = \phi ^{1}(I + \nobreak J)\).
2.1 The geometry of toric fiber products
If \(I \times_{\mathcal{A}}J\) is a codimension zero toric fiber product, the relation between the schemes \(\operatorname{Spec}( \mathbb{K}[x]/I)\), \(\operatorname{Spec}(\mathbb{K}[y]/J)\) and \(\operatorname{Spec}( \mathbb{K}[z]/(I \times_{\mathcal{A}}J))\) can be explained in the language of GIT (geometric invariant theory) quotients. Since I and J are homogeneous with respect to the grading by \(\mathcal{A}\), both \(\operatorname{Spec}( \mathbb{K}[x]/I)\) and \(\operatorname{Spec}(\mathbb{K}[y]/J)\) have an action of a \((\dim\mathcal{A} 1)\)dimensional torus T. Thus the product scheme \(\operatorname{Spec}( \mathbb{K}[x]/I) \times\operatorname{Spec}(\mathbb{K}[y]/J)\) possesses an action of T via t⋅(x,y)=(tx,t ^{−1} y).
Proposition 2.2
Proof
Definition 2.3
With this new definition, Proposition 2.2 is equivalent to the statement:
Proposition 2.4
2.2 Persistence of normality
One of the most basic questions about an ideal I in a ring R is whether or not the quotient R/I is normal. When I is a toric ideal, \(\mathbb{K}[x]/I\) is an affine semigroup ring and normality can be characterized in terms of the semigroup having no holes. In algebraic statistics, normality implies favorable properties of sampling algorithms for contingency tables [4, 36]. In this section we show that normality persists under codimension zero toric fiber products. We only treat the case of (not necessarily toric) prime ideals, which suffices in many situations (see for instance [35, Proposition 2.1.16]).
Theorem 2.5
Let I and J be homogeneous prime \(\mathbb{N}\mathcal{A}\)graded ideals, with \(\mathcal{A}\) linearly independent, and suppose that \(\mathbb{K}[x]/I\) and \(\mathbb{K}[y]/J\) are normal domains (that is, integrally closed in their field of fractions). If \(\mathbb{K}\) is algebraically closed, then \(\mathbb{K} [z]/ (I \times_{\mathcal{A}}J)\) is normal.
The assumption that \(\mathbb{K}\) is algebraically closed is needed to ensure that \(\mathbb{K}[z]/ (I \times_{\mathcal{A}}J)\) is a domain. This holds more generally if I and J are geometrically prime (see Theorem 3.1). If this is given, the field assumption can be weakened to \(\mathbb{K}\) being a perfect field, that is, a field \(\mathbb{K}\) such that either \(\operatorname {char}(\mathbb{K}) = 0\) or \(\operatorname {char}(\mathbb{K}) = p\) and \(\mathbb{K}= \left\lbrace a^{p} : a\in\mathbb{K} \right\rbrace\). The proof of Theorem 2.5 is based on the following observation which is easy and independent of the codimension of \(\mathcal{A}\).
Lemma 2.6
The multigraded Segre product is a direct summand of the tensor product \(R \otimes_{\mathbb{K}} S\) (as a module over the subring).
Proof
The inclusion \(0 \to\bigoplus_{a\in\mathbb{N}\mathcal{A}} R_{a} \otimes_{\mathbb{K}} S_{a} \to \bigoplus_{a\in\mathbb{N}\mathcal{A}}\bigoplus_{b\in\mathbb {N}\mathcal{A}} R_{a} \otimes _{\mathbb{K}} S_{b}\) splits via the \((\bigoplus_{a\in\mathbb{N}\mathcal{A}} R_{a} \otimes_{\mathbb{K}} S_{a})\)module homomorphism that maps \(x^{i}_{j}\otimes y^{l}_{k}\) to itself if a _{ i }=a _{ l } and zero otherwise. □
We anticipate that Lemma 2.6 will be useful in relating properties of multigraded Segre products to those of the factors. For instance, a careful analysis of the Castelnuovo–Mumford regularity would be interesting, but is beyond the scope of this paper. We apply the lemma to prove persistence of normality in codimension zero. Note that the codimension requirement enters because only if \(\mathcal{A}\) is linearly independent, Lemma 2.6 gives us a handle on the toric fiber product.
Proof of Theorem 2.5
Let \(R = \mathbb{K}[x]/I\) and \(S=\mathbb{K}[y]/J\). It is easy to see directly (and also follows from Theorem 3.1 below) that \(\mathbb{K}[z]/(I \times _{\mathcal{A}}J)\) is a domain, given that \(\mathbb{K}\) is algebraically closed. An algebraically closed field is perfect and therefore, if R and S are normal, then \(R\otimes _{\mathbb{K} }S\) is normal. This follows from Serre’s criterion and [38, Theorem 6]. Since a direct summand of a normal domain is normal, Lemma 2.6 completes the proof. □
The main case of interest for our applications is when the ideals I and J are toric ideals and various special cases have been proved in the algebraic statistics literature. For example, Ohsugi [27] proves this for cut ideals, Sullivant [33] for hierarchical models, and Michałek [24] for groupbased phylogenetic models. The proofs of these results are essentially the same, and consists of analyzing a toric fiber product of the grading semigroup. We introduce this setting now.
2.3 Fiber products of vector configurations
3 Persistence of primary decomposition
Primary decompositions of toric fiber products consist of toric fiber products of primary components. To state the result, recall that an ideal is geometrically primary if it is primary over any algebraic extension of the coefficient field.
Theorem 3.1
Proof
For the second claim, since \(I_{i} \times_{\mathcal{A}}J_{j}\) is the inverse image of I _{ i }+J _{ j }, and inverse images of primary ideals are primary, it suffices to show, for any geometrically primary ideals \(I \subseteq\mathbb{K}[x]\) and \(J \subseteq\mathbb{K} [y]\), that \(I+J \subseteq\mathbb{K}[x,y] \) is geometrically primary. First, note that the statement clearly holds if I and J are geometrically prime ideals, since the join of two irreducible varieties is irreducible. The proof of Proposition 1.2 (iv) in [30] contains the cases of geometrically primary ideals. □
Theorem 3.2

there exists \(\mathbf{a}\in\mathbb{N}\mathcal{A}\) such that \((I_{i_{1}})_{\mathbf{a}} \nsubseteq (I_{i_{2}})_{\mathbf{a}}\) and \((J_{j_{2}})_{\mathbf{a}}\neq\mathbb {K}[y]_{\mathbf{a}}\), or

there exists \(\mathbf{b}\in\mathbb{N}\mathcal{A}\) such that \((J_{j_{1}})_{\mathbf{b}} \nsubseteq (J_{j_{2}})_{\mathbf{b}}\) and \((I_{i_{2}})_{\mathbf{b}}\neq\mathbb {K}[x]_{\mathbf{b}}\).
Proof
To deal with redundancy of the decomposition, we must describe conditions on \(I, K \subseteq\mathbb{K}[x]\) and \(J,L \subseteq\mathbb{K}[y]\) that imply \(I \times_{\mathcal{A}}J \subseteq K \times_{\mathcal{A}}L\). Let \(R =\mathbb{K}[x]/I\), \(S = \mathbb{K}[y]/J\), \(R' = \mathbb {K}[x]/K\), and \(S' = \mathbb{K} [y]/L\). Since \(\mathcal{A}\) is linearly independent, the rings \(\mathbb{K}[z]/ (I \times_{\mathcal{A}}J)\) and \(\mathbb{K}[z]/ (K \times_{\mathcal{A}}L)\) are multigraded Segre products. So \(I \times _{\mathcal{A}}J \subseteq K \times_{\mathcal{A}} L\) if and only if \(R' \times_{\mathbb{N}\mathcal{A}}S'\) is a quotient of \(R \times_{\mathbb{N}\mathcal{A}}S\) by the ideal generated by the image of \(K \times_{\mathcal{A}}L\) in \(R \times_{\mathbb {N}\mathcal{A}}S\). On the level of the homogeneous components, we require that \(R'_{\mathbf{a}}\otimes_{\mathbb{K}}S'_{\mathbf{a}}= R_{\mathbf{a}} \otimes_{\mathbb{K}} S_{\mathbf{a}}/ (K \times_{\mathcal{A}}L)_{\mathbf{a}}\), as \(\mathbb {K}\)vector spaces. There are two ways that \(R'_{\mathbf{a}}\otimes_{\mathbb{K}}S'_{\mathbf{a}}\) could be a quotient of \(R_{\mathbf{a}}\otimes _{\mathbb{K}} S_{\mathbf{a}}\). If I _{ a }⊆K _{ a } and J _{ a }⊆L _{ a }, then \((I \times_{\mathcal{A}}J)_{\mathbf{a}} \subseteq(K \times_{\mathcal {A}}L)_{\mathbf{a}}\), in which case we have the desired quotient. The second way is if the tensor product \(R'_{\mathbf{a}}\otimes_{\mathbb{K}} S'_{\mathbf{a}} = \{0 \}\), which happens if and only if either \(R'_{\mathbf{a}}\) or \(S'_{\mathbf{a}}\) is {0}. On the level of ideals, this happens if and only if either \(K_{\mathbf{a}}= \mathbb{K}[x]_{\mathbf{a}}\) or \(L_{\mathbf{a}}= \mathbb{K}[y]_{\mathbf{a}}\).
The decomposition (3) is redundant if and only if there are i _{1},i _{2} and j _{1},j _{2} where \(I_{i_{1}} \times_{\mathcal{A}}J_{j_{1}} \subseteq I_{i_{2}} \times_{\mathcal{A}} J_{j_{2}}\) (where one of i _{1}=i _{2} and j _{1}=j _{2} is allowed, but not both). Now \(I_{i_{1}} \times_{\mathcal{A}}J_{j_{1}} \subseteq I_{i_{2}} \times _{\mathcal{A}}J_{j_{2}}\) if and only if for all \(\mathbf{a}\in\mathbb{N}\mathcal{A}\), \((\mathbb {K}[x]/I_{i_{2}})_{\mathbf{a}}\otimes_{\mathbb{K}}(\mathbb{K} [y]/J_{j_{2}})_{\mathbf{a}}\) is a quotient of \((\mathbb{K}[x]/I_{i_{1}})_{\mathbf{a}}\otimes_{\mathbb{K}}(\mathbb{K} [y]/J_{j_{1}})_{\mathbf{a}}\). This happens if and only if for each \(\mathbf{a}\in\mathbb {N}\mathcal{A}\) the condition in the previous paragraph is satisfied. Thus, \(I_{i_{1}} \times_{\mathcal {A}}J_{j_{1}} \nsubseteq I_{i_{2}} \times_{\mathcal{A}}J_{j_{2}}\) if and only if the negation of this condition holds. Choosing a from the first condition of the theorem with respect to j=j _{2}, yields the desired noncontainment in the case i _{1}≠i _{2}. If i _{1}=i _{2} and j _{1}≠j _{2}, we choose b from the second condition of the theorem with respect to i=i _{1}. This proves the sufficiency of the conditions.
The two conditions are necessary since the first is necessary for \(I_{i_{1}} \times_{\mathcal{A}} J_{j} \nsubseteq I_{i_{2}} \times_{\mathcal{A}}J_{j}\), while the second is necessary for \(I_{i} \times_{\mathcal{A}}J_{j_{1}} \nsubseteq I_{i} \times_{\mathcal{A}}J_{j_{2}}\). □
Corollary 3.3
Proof
We combine Theorems 3.1 and 3.2. Since the ideals I _{ i } and J _{ j } are all geometrically primary, the decomposition of \(I \times_{\mathcal{A}}J\) is a primary decomposition. Since the decomposition of I is irredundant, for each i _{1}≠i _{2} there exists \(\mathbf{a}\in \mathbb{N}\mathcal{A} \) such that \((I_{i_{1}})_{\mathbf{a}}\nsubseteq(I_{i_{2}})_{\mathbf{a}}\) and, by assumption, for all j \((J_{j})_{\mathbf{a}}\neq\mathbb{K}[y]_{\mathbf{a}}\). Similarly, the decomposition of J is irredundant, for each j _{1}≠j _{2} there exists \(\mathbf{b}\in \mathbb{N}\mathcal{A} \) such that \((J_{j_{1}})_{\mathbf{b}}\nsubseteq(J_{j_{2}})_{\mathbf{b}}\) and, by assumption, for all i, \((I_{i})_{\mathbf{b}}\neq\mathbb{K}[y]_{\mathbf{b}}\). This implies that the decomposition is irredundant. □
To apply Corollary 3.3 iteratively, we need to control when its hypotheses are preserved.
Lemma 3.4

for all \(\mathbf{b} \in\mathbb{N}\mathcal{B}\) \((I)_{\mathbf{b}} \neq \mathbb{K}[x]_{\mathbf{b}}\), and

for all \(\mathbf{a}\in\mathbb{N}\mathcal{A}\) \((J)_{\mathbf {a}} \neq\mathbb{K} [y]_{\mathbf{a}}\).
Proof
Example 3.5
(Monomial primary decomposition)
4 Generators of toric fiber products of toric ideals
To each higher codimension toric fiber product there is a natural codimension zero product (Definition 4.1) which contributes many of the generators. There are also additional generators glued from certain pairs of generators of the original ideals. Keeping track of the different contributions requires substantial notation which we found managable only in the case of toric ideals. To verify our results we require that the generating sets of the original ideals satisfy the compatible projection property (Definition 4.7). Any generating set can be extended to one that satisfies this property, but it may be inscrutable how to do so. In special cases, however, the condition becomes clear. For instance, in codimension one toric fiber products the simpler slowvarying condition (Definition 4.10) implies the compatible projection property.
Definition 4.1
The ideal \(\tilde{I} \times_{\tilde{\mathcal{A}}}\tilde{J}\) is the associated codimension zero toric fiber product to \(I \times_{\mathcal{A}}J\).
To describe generators of the toric ideal \(I_{\mathcal{B}\times _{\mathcal{A}}\mathcal{C}}\), we first relate them to Markov bases, via the fundamental theorem [6]. Let \(A \in \mathbb{Z}^{d \times n}\) be a matrix, which defines a toric ideal \(I_{A} = \left\langle p^{\mathbf{u}} p^{\mathbf{v}}: A \mathbf{u}= A \mathbf{v}\right\rangle \subset \mathbb{K}[p_{1},\dots,p_{n}]\). Hence, binomial generators of I _{ A } correspond to elements in kerA. The matrix A defines an \(\mathbb{N}\)linear map \(\mathbb{N}^{n} \to\mathbb{Z}^{d}\) whose image is the affine semigroup \(\mathbb{N}A\). Let \(\mathbf{b}\in\mathbb{N}A\). The fiber of b is the set \(A^{1}[\mathbf{b} ] := \{ \mathbf{u}\in\mathbb{N}^{n} : A \mathbf{u}= \mathbf{b}\}\). Let \(\mathcal{F} \subseteq\ker A\). For each \(\mathbf{b}\in\mathbb{N}A\) we associate a graph \(A^{1}[\mathbf {b}]_{\mathcal{F}}\), with vertex set consisting of all lattice points in A ^{−1}[b] and an edge between u,v∈A ^{−1}[b] if either u−v or \(\mathbf{v} \mathbf{u}\in\mathcal{F} \). A finite subset \(\mathcal{F}\subseteq\ker A\) is a Markov basis of A if the graph \(A^{1}[\mathbf{b}]_{\mathcal{F}}\) is connected for each \(\mathbf {b}\in\mathbb{N}A\). The fundamental theorem of Markov bases connects these latticebased definitions with the generators of the toric ideal I _{ A }.
Theorem 4.2
(Fundamental Theorem of Markov Bases [6])
A finite subset \(\mathcal{F}\subseteq\ker A\) is a Markov basis of A if and only if the set of binomials \(\{ p^{\mathbf{f}^{+}}  p^{\mathbf{f}^{}} : \mathbf{f}\in\mathcal{F}\}\) generates I _{ A }.
4.1 Codimension zero toric fiber products
Theorem 4.3
(Codimension zero toric fiber products, [34])
4.2 The compatible projection property
Define \(L(w^{\mathbf{v}_{2}})\) to be the set of all monomials x ^{ v } in \(\mathbb{K} [x]\) such that \(\phi_{xw}(x^{\mathbf{v}}) = w^{\mathbf{v}_{2}}\). Similarly, define \(R(w^{\mathbf{v} _{1}})\) to be the set of monomials y ^{ v } in \(\mathbb{K}[y]\) such that \(\phi _{yw}(y^{\mathbf{v}}) = w^{\mathbf{v}_{1}}\). By construction if \(x^{\mathbf{v}} \in L(w^{\mathbf{v}_{2}})\) and \(y^{\mathbf{v}'} \in R(w^{\mathbf{v}_{1}})\) then x ^{ v } f and y ^{ v′} g, when written as tableaux and after reordering rows, have exactly the same first column. Thus, we can form the binomial \(\operatorname {glue}(x^{\mathbf{v}}f, y^{\mathbf{v}'} g)\).
Definition 4.4
Proposition 4.5
Proof
Consider the natural \(\mathbb{N}\)linear projection maps \(\gamma: \mathbb{N} ^{\mathcal{B} \times_{\mathcal{A}}\mathcal{C}} \to\mathbb{N}^{r}, \gamma(e^{i}_{jk}) = e_{i}\), \(\gamma_{1} : \mathbb {N}^{\mathcal{B}} \rightarrow \mathbb{N}^{r}, \gamma_{1}( e^{i}_{j}) = e_{i}\), and \(\gamma_{2}: \mathbb{N}^{\mathcal{C}}\rightarrow\mathbb{N}^{r}, \gamma_{2}(e^{i}_{k}) = e_{i}\). These projections evaluate the additional multidegrees appearing in the definition of the associated codimension zero product. They are also defined on the fibers \(\mathcal{B}^{1}[\mathbf{b}]\) and \(\mathcal{C}^{1}[\mathbf{c}]\) and the graphs \(\mathcal{B} ^{1}[\mathbf{b}]_{\mathcal{F}}\) and \(\mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}}\). Note that if \(\mathbf{f}\in\ker\mathcal{B}\) then \(\gamma_{1}(\mathbf{f}) \in\ker\mathcal{A}\), and similarly for γ, and γ _{2}.
Definition 4.6
Let \(\mathcal{F}\subseteq\ker\mathcal{B}\). The graph \(\gamma_{1}( \mathcal{B} ^{1}[\mathbf{b} ]_{\mathcal{F}})\) has vertex set \(\gamma_{1}(\mathcal{B}^{1}[\mathbf{b}])\) and an edge between u′ and v′ if there are \(\mathbf{u}, \mathbf{v}\in\mathcal {B}^{1}[\mathbf{b}]\) such that u and v are connected by an edge in \(\mathcal{B}^{1}[\mathbf {b}]_{\mathcal{F}}\) and γ _{1}(u)=u′ and γ _{1}(v)=v′. Similarly define the graphs \(\gamma_{2}( \mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}})\) and \(\gamma((\mathcal {B}\times_{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b},\mathbf{c})]_{\mathcal{H}})\) where \(\mathcal{G}\subseteq \ker\mathcal{C}\) and \(\mathcal{H} \subseteq \ker\mathcal{B}\times_{\mathcal{A}}\mathcal{C}\). These are the projection graphs.
Given two graphs G and H with overlapping vertex sets, their intersection G∩H is the graph with vertex set V(G)∩V(H) and edge set E(G)∩E(H).
Definition 4.7
The next lemma is the main technical result allowing us to produce generating sets for toric fiber products.
Lemma 4.8
Proof
 1.
\(V(\gamma( (\mathcal{B}\times_{\mathcal{A}}\mathcal {C})^{1}[ (\mathbf{b}, \mathbf{c})]_{{\operatorname {\mathbf {Glue}}}(\mathcal{F} ,\mathcal{G})})) = V(\gamma_{1}( \mathcal{B}^{1}[\mathbf {b}]_{\mathcal{F}}) ) \cap V(\gamma_{2}( \mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}}))\),
 2.
\(E(\gamma( (\mathcal{B}\times_{\mathcal{A}}\mathcal {C})^{1}[ (\mathbf{b}, \mathbf{c})]_{{\operatorname {\mathbf {Glue}}}(\mathcal{F} ,\mathcal{G})})) = E( \gamma_{1}( \mathcal{B}^{1}[\mathbf {b}]_{\mathcal{F}}) ) \cap E(\gamma_{2}( \mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}}))\).
Proof of part (1) We must show that if d is in both \(\gamma_{1}( \mathcal {B}^{1}[\mathbf{b} ]_{\mathcal{F}} )\) and \(\gamma_{2}( \mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}})\) then \(\mathbf {d}\in\gamma( (\mathcal{B} \times_{\mathcal{A}} \mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{{\operatorname {\mathbf {Glue}}}(\mathcal {F},\mathcal{G})})\). By assumption there are \(\mathbf{u}_{1} \in\mathcal{B}^{1}[\mathbf{b}]\) and \(\mathbf{u}_{2} \in\mathcal{C}^{1}[\mathbf{c}]\) such that γ _{1}(u _{1})=γ _{2}(u _{2})=d. Since π _{1}(b)=π _{2}(c) and γ _{1}(u _{1})=γ _{2}(u _{2}) the corresponding monomials \(x^{\mathbf{u}_{1}}\) and \(y^{\mathbf{u}_{2}}\) have the same \(\tilde{\mathcal{A}}\) degree. Since \(\tilde {\mathcal{A}}\) is linearly independent, the monomial \(x^{\mathbf{u}_{1}} y^{\mathbf{u}_{2}} \in\mathbb{K}[x] \otimes_{\mathbb{K}} \mathbb{K}[y]\) is in the image of \(\phi_{I_{\tilde{\mathcal{B}}}, J_{\tilde {\mathcal{C} }}}\). Let z ^{ u } be a monomial such that \((\tilde{\mathcal{B}} \times _{\tilde{\mathcal{A}}}\tilde{\mathcal{C} })\mathbf{u}= (\mathbf{b}, \mathbf{c}, \mathbf{d})\) and hence \((\mathcal{B}\times_{\mathcal {A}}\mathcal{C})\mathbf{u}= (\mathbf{b}, \mathbf{c})\). But this implies \(\mathbf{d}\in\gamma( (\mathcal{B}\times_{\mathcal{A}}\mathcal {C})^{1}[ (\mathbf{b}, \mathbf{c})]_{{\operatorname {\mathbf {Glue}}}(\mathcal{F} ,\mathcal{G})})\).
Proof of part (2) Suppose that d and e are both in \(\gamma_{1}( \mathcal{B} ^{1}[\mathbf{b} ]_{\mathcal{F}}) \) and \(\gamma_{2}( \mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}})\), and they are connected by an edge. We must show that d and e are connected by an edge in \(\gamma( (\mathcal{B}\times_{\mathcal{A}} \mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{{\operatorname {\mathbf {Glue}}}(\mathcal {F},\mathcal{G})})\). To do this, we must show that there are w _{1} and \(\mathbf{w}_{2} \in(\mathcal {B}\times_{\mathcal{A}}\mathcal{C} )^{1}[(\mathbf{b}, \mathbf{c})]\), with γ(w _{1})=d and γ(w _{2})=e such that \(\mathbf{w}_{1}  \mathbf{w}_{2} \in{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G})\).
Since there is an edge in \(\gamma_{1}( \mathcal{B}^{1}[\mathbf {b}]_{\mathcal{F}})\) between d and e, there exist u _{1} and u _{2} in \(\mathcal{B}^{1}[\mathbf{b}]\) such that γ _{1}(u _{1})=d, γ _{1}(u _{2})=e and \(\mathbf{u}_{1}  \mathbf{u} _{2} = \mathbf{f}\in \mathcal{F}\). Similarly, there are v _{1} and \(\mathbf{v}_{2} \in\mathcal{C} ^{1}[\mathbf{c}]\) such that γ _{2}(v _{1})=d, γ _{2}(v _{2})=e and \(\mathbf{v}_{1}  \mathbf{v} _{2} = \mathbf{g} \in\mathcal{G}\). By part (1), there exists \(\mathbf{w}_{1} \in (\mathcal{B}\times_{\mathcal{A}}\mathcal{C} )^{1}[(\mathbf{b}, \mathbf{c})] \) which projects to (u _{1},v _{1}) and \(\mathbf{w}_{2} \in (\mathcal{B} \times_{\mathcal{A}} \mathcal{C})^{1}[(\mathbf{b}, \mathbf{c})]\) which projects to (u _{2},v _{2}). There are many choices for w _{1} and w _{2}. We claim that we can choose them so that \(\mathbf{w}_{1}  \mathbf{w}_{2} \in{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal {G})\), which completes the proof.
The first block of rows in both u _{1}−u _{2} and v _{1}−v _{2}, have the same I _{1} and \(I_{1}'\) because these blocks correspond to the common binomial \((w^{\mathbf{s}_{1}}  w^{\mathbf{s}_{2}})\) in \(\phi_{xw}(f) = w^{\mathbf{r} _{1}}(w^{\mathbf{s}_{1}}  w^{\mathbf{s}_{2}})\) and \(\phi_{yw}(g) = w^{\mathbf{r}_{2}}(w^{\mathbf {s}_{1}}  w^{\mathbf{s}_{2}})\). Note that this corresponds to d−e. This implies that in the second and third blocks of rows of u _{1} and of u _{2} we have exactly the same multisets of indices in the first column. This explains why I _{2} and I _{3} appear in both the u _{1} and the u _{2} tableaux. A similar argument shows that \(I_{2}^{*}\) and \(I_{3}^{*}\) should appear in both v _{1} and v _{2}. Finally, we must have that the multiset of indices that appear in I _{2} and I _{3} together equals the multiset of indices that appear in \(I_{2}^{*}\) and \(I_{3}^{*}\) together. By our usual assumption that \(\gcd( w^{\mathbf{r}_{1}}, w^{\mathbf{r} _{2}}) = 1\), we see that the multisets I _{2} and \(I_{2}^{*}\) are disjoint. This implies that, as multisets, \(I_{2} \subseteq I_{3}^{*}\) and \(I_{2}^{*} \subseteq I_{3}\).
Theorem 4.9
Let \(\mathcal{H}\subset\ker\tilde{\mathcal{B}}\times_{\tilde {\mathcal{A}}}\tilde{\mathcal{C}}\) be a Markov basis for the associated codimension zero toric fiber product. Let \(\mathcal{F} \subseteq\ker \mathcal{B}\) and \(\mathcal{G}\subseteq\ker\mathcal{C}\). Then \(\mathcal{H}\cup{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G})\) is a Markov basis for \(\mathcal{B}\times_{\mathcal {A}}\mathcal{C}\) if and only if \(\mathcal{F} \) and \(\mathcal{G}\) have the compatible projection property.
Proof
We must show that for any \((\mathbf{b}, \mathbf{c}) \in\mathbb{N}( \mathcal{B}\times_{\mathcal{A}}\mathcal{C})\) the graph \((\mathcal{B}\times_{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{ \mathcal{H}\cup{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G})}\) is connected. For each \(\mathbf{d}\in\mathbb{N}\mathcal{D}\) consider the subgraph of \((\mathcal{B} \times_{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{ \mathcal{H}\cup{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G} )}\) whose vertices consist of all \((\mathbf{u}, \mathbf{v}) \in(\mathcal {B}\times_{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]\) such that γ _{1}(u)=γ _{2}(v)=d. This is precisely the set \(\tilde{\mathcal{B}} \times_{\tilde{\mathcal{A}}}\tilde {\mathcal{C}}^{1}[ (\mathbf{b}, \mathbf{c}, \mathbf{d} )]\). This subgraph is connected since \(\mathcal{H}\) is a Markov basis for \(\tilde {\mathcal{B}} \times_{\tilde{\mathcal{A}}} \tilde{\mathcal{C}}\). The graph \(\gamma( (\mathcal{B}\times _{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{ \mathcal{H}\cup \operatorname {\mathbf {Glue}}(\mathcal{F}, \mathcal{G})})\) equals the graph \(\gamma( (\mathcal{B} \times_{\mathcal{A}} \mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{\operatorname {\mathbf {Glue}}(\mathcal{F}, \mathcal{G})})\) because \(\mathcal{H}\) is contained in the kernel of the projection γ. This graph is connected since \(\mathcal{F}\) and \(\mathcal{G}\) have the compatible projection property and by Lemma 4.8. But if the image of a map of graphs is connected and each fiber is connected, then the graph itself is connected, which completes the proof of the if direction.
Conversely, if every fiber is connected, the graph \((\mathcal{B}\times _{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{ \mathcal{H}\cup{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G})}\) is connected, so the graph \(\gamma( (\mathcal{B}\times_{\mathcal{A}}\mathcal{C})^{1}[ (\mathbf{b}, \mathbf{c})]_{\operatorname {\mathbf {Glue}}(\mathcal{F}, \mathcal{G} )})\) is connected. By Lemma 4.8, this equals \(\gamma_{1}( \mathcal{B}^{1}[\mathbf{b}]_{\mathcal{F}}) \cap\gamma_{2}( \mathcal {C}^{1}[\mathbf{c}]_{\mathcal{G}})\) so that \(\mathcal{F}\) and \(\mathcal{G}\) have the compatible projection property. □
Theorem 4.9 gives an explicit way to construct a Markov basis for \(\mathcal{B} \times_{\mathcal{A}}\mathcal{C}\). However, there remains a serious difficulty in finding sets \(\mathcal{F} \subset\ker\mathcal{B}\) and \(\mathcal{G}\subset\ker\mathcal{C}\) which have the compatible projection property. In general, it is not true that \(\mathcal{F}\) and \(\mathcal{G} \) can be arbitrary Markov bases of \(\mathcal{B}\) and \(\mathcal{C}\).
4.3 Slowvarying Markov bases
In the remainder of the section, we describe the slowvarying condition (generalizing [11]) which, if the codimension is one, can be used to show that a given pair of Markov bases satisfies the compatible projection property.
Definition 4.10
Suppose that \(\mathcal{B}\times_{\mathcal{A}}\mathcal{C}\) is a codimension one toric fiber product. Let \(\mathbf{h}\in\mathbb{Z}^{r}\) be nonzero. Let \(\mathcal {F}\subseteq\ker \mathcal{B}\) and \(\mathcal{G}\subseteq\ker\mathcal{C}\). Then \(\mathcal{F}\) and \(\mathcal{G}\) are slowvarying with respect to h if for all \(\mathbf {f}\in\mathcal{F}\), γ _{1}(f)=0, or ±h; and for all \(\mathbf{g}\in\mathcal{G}\), γ _{2}(g)=0 or ±h.
Proposition 4.11
Let h generate \(\ker\mathcal{A}\). If the maximum 1norm of any element in \(\mathcal{F}\) or \(\mathcal{G}\) is less than 2∥h∥_{1}, then \(\mathcal{F}\) and \(\mathcal{G}\) are slowvarying with respect to h.
Proof
Since γ _{1}(f) must be a multiple of h and ∥γ _{1}(f)∥_{1}≤∥f∥_{1}, if ∥f∥_{1}<2∥h∥_{1} then γ _{1}(f) is either 0 or ±h. A similar statement holds for γ _{2}(g). □
Theorem 4.12
Suppose that \(\mathcal{B}\times_{\mathcal{A}}\mathcal{C}\) is a codimension one toric fiber product. Let \(\mathcal{H}\) be a Markov basis for \(\tilde{\mathcal{B}} \times _{\tilde{\mathcal{A}}}\tilde{\mathcal{C}}\). Let \(\mathcal{F}\) and \(\mathcal{G}\) be Markov bases for \(\mathcal{B}\) and \(\mathcal {C}\) that are slowvarying with respect to \(\mathbf{h}\in\ker\mathcal{A}\). Then \(\mathcal{H}\cup \operatorname {\mathbf {Glue}}(\mathcal{F}, \mathcal{G} )\) is a Markov basis for \(\mathcal{B}\times_{\mathcal{A}}\mathcal{C}\).
Proof
Since the toric fiber product is codimension one, the vertex sets of the graphs \(\gamma_{1}(\mathcal{B}^{1}[\mathbf{b}]_{\mathcal{F}})\) and \(\gamma _{2}( \mathcal{C}^{1}[\mathbf{c} ]_{\mathcal{G}})\) are subsets of the lattice \(\mathbb{Z}\mathbf{h}\). Since \(\mathcal{F}\) and \(\mathcal{G}\) are Markov bases, these graphs are connected. By the slowvarying condition, the edges connect two points whose difference is ±h. Hence the graphs \(\gamma_{1}( \mathcal{B} ^{1}[\mathbf{b} ]_{\mathcal{F}})\) and \(\gamma_{2}( \mathcal{C}^{1}[\mathbf{c}]_{\mathcal{G}})\) are intervals of ordered points. The intersection of two such graphs is another graph of the same type, and is also connected. Thus \(\mathcal{F}\) and \(\mathcal{G}\) have the compatible projection property and Theorem 4.9 then implies that \(\mathcal{H}\cup{ \operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G})\) is a Markov basis for \(\mathcal{B}\times_{\mathcal{A}}\mathcal{C}\). □
5 Application: Markov bases for hierarchical models
Let Γ be a simplicial complex with vertex set V, and let \(d \in \mathbb{Z}^{V}_{\geq2}\) a vector of integers. These data define a hierarchical model as in Sect. 1.1, and hence a toric ideal I _{ Γ,d }. For any homogeneous ideal I, let μ(I) denote the largest degree of a minimal generator of I, which is an invariant of the ideal. This is a coarse measure of the complexity of the ideal I. If Γ is a graph and d _{ v }=2 for all v∈V, μ(I _{ Γ,d }), is an invariant of Γ dubbed the Markov width in [5]. We calculate μ(Γ,d):=μ(I _{ Γ,d }) for certain simplicial complexes Γ and vectors d. The results of Sect. 4 are also useful to explicitly construct Markov bases of these hierarchical models.
Proposition 5.1
Proof
Proposition 5.2
Proof
5.1 Small examples
Lemma 5.3
 (1)Let S _{ V }=2^{ V }∖{V}, be the boundary of a (#V−1)dimensional simplex. Then \(I_{S_{V}, {\bf2}}\) is generated by a single binomial:$$\prod_{i \in\mathrm{D}_{V}: \ i \_{1} \mathrm{even}} p_{i}  \prod _{i \in\mathrm{D} _{V}: \ i \_{1} \mathrm{odd}} p_{i}. $$
 (2)Let Γ be a simplicial complex on V, letbe the cone over Γ with apex v, and let \(\mathcal{F}\) be a (minimal) generating set of I _{ Γ,d }. Then \(I_{{\rm cone}_{v}(\varGamma), d_{V \cup\{v\}}}\) is (minimally) generated by$${\rm cone}_{v}(\varGamma) = \varGamma\cup\bigl\{F \cup\{v\} : F \in\varGamma \bigr\} $$
Proof
(1) According to the dimension formula (4), \(I_{S_{V}, {\bf 2}}\) is generated by a single equation. The proof of (4) in [16] shows that the given binomial generates the ideal.
(2) This follows because one can rearrange the rows and columns of \(A_{{\rm cone}_{v}(\varGamma), d_{V \cup\{v\}}}\) so that it is a block diagonal matrix with d _{ v } diagonal blocks with the matrix A _{ Γ,d } along the diagonal. This decomposition appears in [17]. □
Example 5.4
(Binary fourcycle)
Similar arguments and the description of Markov bases of small cycles in Lemma 5.9 can be used to get an explicit description of Markov bases of the fourcycles that appear in Example 1.2. We can also produce analogous results for higher dimensional complexes.
Theorem 5.5
Let B _{ n } be the simplicial complex with vertex set [n+2] and minimal nonfaces [n] and {n+1,n+2}. The ideal \(I_{B_{n}, {\bf2}}\) has a generating set consisting of binomials of degrees 2,2^{ n−1}, and 2^{ n }.
Proof
For i=1,2 let Γ _{ i } be the cone B _{ n }∖(n+i) over the boundary of the simplex on [n]. Then B _{ n }=Γ _{1}∪Γ _{2}. According to part (1) of Lemma 5.3 the Markov basis of \(I_{\varGamma_{1} \cap \varGamma_{2} , {\bf2}}\) consists of a single element of degree 2^{ n−1}; and according to part (2) the ideals \(I_{\varGamma_{i}, {\bf2}}\) are each generated by two binomials of degree 2^{ n−1}. Since 2^{ n−1}<2×2^{ n−1}, by Proposition 4.11, the Markov bases for \(I_{\varGamma_{1}, {\bf2}}\) and \(I_{\varGamma_{2}, {\bf2}}\) are slowvarying with respect to the Markov basis of \(I_{\varGamma_{1} \cap\varGamma_{2} , {\bf2}}\). The set of glue moves consists of 4 binomials of degree 2^{ n−1}.
The simplicial complex \(\tilde{\varGamma}\) appearing in the associated codimension zero toric fiber product has [n] as an additional face. Consequently it consists of the boundaries of two (n−1)dimensional simplices that share a single facet. By part (1) of Lemma 5.3, the Markov basis of the boundary of an (n−1)dimensional simplex consists of a single element of degree 2^{ n }. The lifting operation preserves degree and produces \(2^{2^{n}}\) elements per boundary simplex, for a total of \(2^{2^{n} +1}\) elements of degree 2^{ n }. Finally, there are 2^{ n } quadrics in \({\rm Quad}\). Theorem 4.12 shows that the union of all these elements is a Markov basis. □
The simplicial complex B _{ n } is the boundary of the polytope that is a bipyramid over a simplex. In particular, it is a simplicial sphere. Theorem 5.5 and the results of [28] provide evidence for the following conjecture.
Conjecture 5.6
Let Γ be a triangulation of a sphere of dimension n. Then the Markov basis of \(I_{\varGamma, {\bf2}}\) consists of elements of degree at most 2^{ n+1}.
To conclude this section, we give an example which shows how the gluing operation can produce Markov basis elements of larger degree than either of the constituent binomials.
Example 5.7
5.2 Cycles and ring graphs
In this subsection, and the next, Γ=G is a graph. We start with cycles and graphs that can be easily constructed from cycles, then explore K _{4}minor free graphs, providing a new proof of the main result in [21]. To set up induction we provide the Markov bases of simple graphs.
Lemma 5.8
Let P be a path and \(d\in\mathbb{Z}^{V}_{\geq2}\) arbitrary, then μ(P,d)=2.
Proof
This follows from Theorem 4.3 or the results on decomposable simplicial complexes in [8, 37]. □
Lemma 5.9
(Small Graphs)
 (1)Let K _{3} be the triangle. The following table contains known values of μ(K _{3},d):$$\begin{array}{c@{\quad }@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} d_1 & 2 & 3 & 3 & 3 & 3 & 4 \\ d_2 & p & 3 & 3 & 3 & 4 & 4 \\ d_3 & q & 3 & 4 & q \geq5 & 4 & 4 \\ \hline \mu(I_{K_{3},d}) & \min(2p,2q) & 6 & 8 & 10 & 12 & 14 \ \\ \end{array} $$
 (2)If C is a fourcycle with edges 12,23,34,41, then μ(C,d) takes the following values:$$\begin{array}{c@{\quad }@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} d_1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 3 \\ d_2 & 2 & 2 & 2 & 2 & 2 & 2 & 3 & 3\\ d_3 & 3 & 3 & 3 & 4 & 4 & 5 & 3 & 3\\ d_4 & 3 & 4 & 5 & 4 & 5 & 5 & 3 & 3\\ \hline \mu(C,d) & 6 & 6 & 6 & 8 & 8 & 10 & 6 & 6 \\ \end{array} $$
 (3)
If C is a fivecycle with edges 12,23,34,45,51, d _{1}=d _{2}=2, and d _{3}=d _{4}=d _{5}=3, then μ(C,d)=6.
 (4)
Let K _{2,3} be the complete bipartite graph on {1,2} and {3,4,5}. If d _{1}=d _{2}=3 and d _{3}=d _{4}=d _{5}=2, then μ(K _{2,3},d)=6.
 (5)
The complete graph K _{4} with \(d = {\bf2}\) satisfies μ(K _{4},2)=6.
Proof
The computation for K _{3} with d=(2,p,q) is contained in the original work of Diaconis and Sturmfels [6]. The values for d=(3,3,q) were determined by Aoki and Takemura [2]. All other values have been computed using 4ti2 and Markov bases are available on the Markov Basis Database [19]. □
Lemma 5.10
Proof
In either case I _{ G,d } is a codimension zero toric fiber product and Theorem 4.3 applies. The statement also follows from results on reducible hierarchical model in [9, 16, 34]. □
Lemma 5.11
Proof
Lemma 5.12
 (1)
If C contains no edge uv with d _{ u },d _{ v }>2 then μ(C,d)=4.
 (2)
If all d _{ v }≤3 and C contains no path u _{1} u _{2} u _{3} u _{4} with all \(d_{u_{i}}>2\), then μ(C,d)≤6.
 (3)
If all d _{ v }≤4 and C contains no path u _{1} u _{2} u _{3} with all \(d_{u_{i}}>2\), then μ(C,d)≤8.
 (4)
If all d _{ v }≤5 and C contains no path u _{1} u _{2} u _{3} with all \(d_{u_{i}}>2\), then μ(C,d)≤10.
Proof
We give a detailed proof of (1). According to Lemma 5.9 the statement holds for cycles of length three. We proceed by induction on the length of C. There are always two nonadjacent vertices u and v in C with d _{ u }=d _{ v }=2. Let V _{1} be the set of vertices on one of the paths in C from u to v, and let V _{2} be the set of vertices on the other path. According to Lemma 5.8 the Markov width of paths is two. By induction we find \(\mu(G_{V_{1}}\cup\, uv, d_{V_{1}}) = 4\) and \(\mu(G_{V_{2}} \cup\, uv, d_{V_{2}}) = 4\), since those graphs are shorter cycles than C satisfying the conditions in (1). By Lemma 5.11, the Markov width of μ(C,d)=4. Statements (2)–(4) follow by the same inductive argument and reducing to the small graphs in Lemma 5.9. □
Cycles can be patched together to form larger graph classes, for example ring graphs.
Definition 5.13
A ring graph is a graph that can be recursively constructed from paths and cycles by disjoint unions, identifying a vertex of disjoint components, and identifying edges on disjoint components. An outerplanar graph is a graph with a planar embedding such that all vertices are on a circle.
Outerplanar graphs are also characterized as the largest minor closed class that excludes K _{4} and K _{2,3}. This in particular implies that all outerplanar graphs are seriesparallel since they have no K _{4}minors. It is easy to see that outerplanar graphs are ring graphs. Recall that a graph is kconnected if there is no way to disconnect it by removing at most k−1 vertices. We need to describe how to decompose 2connected ring graphs into cycles.
Definition 5.14

the union of all C _{ i } is G, and

the intersection of C _{1}∪⋯∪C _{ i } and C _{ i+1} is an edge for 1≤i<k.
Any 2connected ring graph must have a cycle decomposition, since a 2connected ring graph is obtained by only identifying edges in disjoint components.
Theorem 5.15
 (1)
there is no edge uv in C with d _{ u },d _{ v }>2 then μ(G,d)≤4;
 (2)
all d _{ v }≤3 and there is no path u _{1} u _{2} u _{3} u _{4} in C with all \(d_{u_{i}}>2\), then μ(G,d)≤6;
 (3)
all d _{ v }≤4 and there is no path u _{1} u _{2} u _{3} in C with all \(d_{u_{i}}>2\), then μ(G,d)≤8;
 (4)
all d _{ v }≤5 and there is no path u _{1} u _{2} u _{3} in C with all \(d_{u_{i}}>2\), then μ(G,d)≤10.
Proof
This follows directly from Lemma 5.8, Lemma 5.10, and Lemma 5.12. □
Definition 5.16
A graph G is Markov slim, if for every independent set I of G the model with d _{ v }≥2 for v∈I and d _{ v }=2 for v∈V(G)∖I has Markov width at most four.
Theorem 5.17
The maximal minorclosed class of Markov slim graphs is the outerplanar graphs.
Proof
By Theorem 5.15 the outerplanar graphs are Markov slim since they are ring graphs. Say that there is a minor closed class larger than the outerplanar graphs, in which every graph is Markov slim. Then this class either contains K _{4} or K _{2,3}. By parts (4) and (5) of Lemma 5.9 neither K _{4} nor K _{2,3} are Markov slim. □
Repeated toric fiber products of cycles reduce computations of the Markov width to the three cycle. Therefore the following conjecture seems natural.
Conjecture 5.18
Our results so far only work with codimension one toric fiber products, which do not raise the degree of generators in the cycle case, and hence we always glued paths at a pair of vertices u,v where d _{ u }=d _{ v }=2. It is not clear whether or not this remains true for larger values of d _{ u },d _{ v }.
5.3 Binary seriesparallel graphs
To prove Theorem 1.1 we apply a classical decomposition of K _{4}minor free graphs.
Definition 5.19

Each graph G∈SP has two distinguished vertices, the top and the bottom vertex, which are different.

The graph K _{2} is in SP.
 If G _{1} and G _{2} are in SP with tops and bottoms t _{1},t _{2}, b _{1},b _{2}, respectively, then
 Series construction

the graph obtained from G _{1} and G _{2} by identifying t _{1} and b _{2} and calling b _{1} and t _{2} the new bottom and top also belongs to SP;
 Parallel construction

the graph obtained form G _{1} and G _{2} by identifying t _{1} and t _{2} and b _{1} and b _{2} (and calling these the new top and bottom) is also in SP.
In a graph without K _{4}minors, every 2connected component is a seriesparallel graph (see [7, Chap. 7]). Since gluing two graphs at a vertex is a codimension zero toric fiber product, to prove Theorem 1.1, we can restrict to seriesparallel graphs. One tool is the following lemma about choices that can be made in the parallel construction.
Lemma 5.20
Suppose that G∈SP has at least four vertices. Then G can be obtained by series or parallel construction from two graphs G _{1} and G _{2} each with fewer vertices than G.
Proof
The series construction of G _{1} and G _{2} clearly produces a graph G with a larger number of vertices. For the parallel construction, if both G _{1} and G _{2} are not single edges then their parallel construction has more vertices than either G _{1} or G _{2}. The only nontrivial case is when one of the two graphs, say G _{1}, is a single edge.
We can assume G _{2} is neither a path of one or two edges, nor K _{3}, since then the resulting graph would have less than three vertices. The graph G _{2} is obtained either by a series or by a parallel construction from two graphs G _{3} and G _{4}. In the case of a parallel construction, consider new graphs \(\tilde{G_{3}}\) and \(\tilde{G_{4}}\) with an edge glued in from t to b in both cases. The resulting parallel construction of \(\tilde{G_{3}}\) and \(\tilde{G_{4}}\) gives the same graph as the parallel construction of G _{1} and G _{2}. In the case of a series construction, one of the graphs G _{3} or G _{4} has ≥3 vertices. Assume that graph is G _{4}. A series construction of G _{1} with G _{3} followed by a parallel construction of the result with G _{4} gives the original graph. We may have to rearrange the tops and bottoms during this construction, but doing so does not change the property of being a seriesparallel graph. □
Theorem 5.21
 (1)
Degree four binomials whose terms have the same degree on the bt subcomplex.
 (2)
Degree two binomials that are slowvarying on the bt subcomplex.
Proof
We proceed by induction on the number of vertices of the graph. The statement is trivially true for connected seriesparallel graphs with one or two vertices, since they have empty Markov basis. There are two graphs with three vertices to consider. For the triangle \(I_{K_{3}, {\bf2}}\) there is one degree four generator and it must project to the zero polynomial along the bt edge, since that edge belongs to K _{3}. In the case of the path with three vertices, there are two quadratic generators, which are slowvarying by Proposition 4.11.
Now let G be a seriesparallel graph with at least four vertices. By Lemma 5.20 it can be built from two graphs G _{1} and G _{2} with strictly smaller numbers of vertices by either a series or a parallel construction. We must show that properties (1) and (2) of the Markov basis are preserved under either of these constructions.
 Lift 1

lifting generators from \(I_{G_{1}, {\bf2}}\) while being constant on G _{2};
 Lift 2

lifting generators from \(I_{G_{2}, {\bf2}}\) while being constant on G _{1};
 Quad

quadratic moves.
If G is obtained from a parallel construction of G _{1} and G _{2}, then the top and bottom vertices can be adjacent or not. If they are adjacent, then we are gluing along an edge. All generators of \(I_{G_{1}, {\bf2}}\) and \(I_{G_{2}, {\bf 2}}\) project to zero along this edge by properties of the lift operation. If the special vertices are not adjacent, we have a codimension one toric fiber product. The associated codimension zero product consists of seriesparallel graphs with fewer vertices. By the argument in the preceding paragraphs, all Markov basis elements obtained from the associated codimension zero toric fiber product satisfies either (1) or (2). Finally, consider \({\operatorname {\mathbf {Glue}}}(\mathcal{F}, \mathcal{G})\). Since all Markov bases satisfy (1) and (2), we only ever glue quadrics, producing more quadrics, which are slowvarying by Proposition 4.11. □
Instead of using binary variables for the triangle in the proof, one could have used larger values of d _{ v } on the vertex of the triangle that is never involved in gluing or identification. This would have given an alternative but less descriptive proof of Theorem 5.15. The procedure yields a larger class than ring graphs, but it is not true that larger d _{ v } on independent sets always produce Markov width four, as illustrated earlier by the fact that K _{2,3} is not Markov slim.
There are further applications of higher codimension toric fiber products in algebraic statistics lurking. For example, ideals of graph homomorphisms [12] generalize classes of toric ideals in algebraic statistics. Given graphs G and H, potentially with loops, the ideal of graph homomorphisms from G to H is I _{ G→H }. In this language, binary hierarchical models arise as the special case where \(H = K_{2}^{o}\) is the complete graph with loops. If H is an edge with one loop, then the homomorphisms from G to H correspond to the independent sets of G. It is known that I _{ G→H } is quadratically generated if G is bipartite, or becomes bipartite after the removal of one vertex [12]. Using Theorem 5.21 as a template, one derives that I _{ G→H } is quadratically generated for seriesparallel G.
Some toric ideals are not toric fiber products themselves, but project to one. With control over the projection one may be able to find a Markov basis anyway. An Example is Norén’s proof of a conjecture by Haws, Martin del Campo, Takemura, and Yoshida [26].
6 Application: conditional independence ideals
A basic problem in the algebraic study of conditional independence is to understand primary decompositions of CIideals. For instance, if a conditional independence model comes from a graph, the minimal primes provide information about families of probability distributions that satisfy the conditional independence constraints but do not factorize according to the graph. Moreover, primary decompositions can provide information about the connectivity of random walks using Markov subbases [20].
In this section J is the generic letter denoting an ideal. This is to avoid confusion between the ideals I _{ G } of Sect. 5 and the CIideals J _{ G } in Sect. 6.2. The results in this section are independent of d=(d _{ v })_{ v∈V }, the vector of cardinalities. It is fixed arbitrarily and does not appear in the notation.
We first develop a general theory for arbitrary conditional independence models. Then we apply it to global Markov ideals of graphs, showing that they are toric fiber products if the graph has a decomposition along a clique.
In statistics one is usually not interested in all of the variety of a CIideal, but only its intersection with the set of probability distributions. The following properties of CIideals imply wellknown properties of conditional independence.
Proposition 6.1

Open image in new window (symmetry);

Open image in new window (decomposition);

Open image in new window (weak union).
6.1 Toric fiber products of CImodels
Let \(\mathcal{M}_{1}, \mathcal{M}_{2}\) be conditional independence models on two (not necessarily disjoint) sets of variables V _{1},V _{2}, respectively. The CIideals \(J_{\mathcal{M}_{1}}\) and \(J_{\mathcal{M}_{2}}\) live in polynomial rings with variables indexed by \(\mathrm{D}_{V_{1}}\), and \(\mathrm{D}_{V_{2}}\), respectively. Their toric fiber product is again a CIideal when certain conditions are satisfied. Our aim is to define the toric fiber product of \(J_{\mathcal{M}_{1}}\) and \(J_{\mathcal{M}_{2}}\) combinatorially, using CIstatements.
Definition 6.2
(The Sgrading)
Let S⊂V. The grading on the polynomial ring \(\mathbb{K}[p_{i} : i \in \mathrm{D}_{V}]\) given by \(\deg(p_{i}) = e_{i_{S}} \in\mathbb {Z}^{\mathrm{D}_{S}}\) is the Sgrading. The conditional independence model \(\mathcal{M}\) is Shomogeneous if each statement Open image in new window in \(\mathcal{M}\) satisfies either S⊆A∪C or S⊆B∪C.
Lemma 6.3
If \(\mathcal{M}\) is Shomogeneous then \(J_{\mathcal{M}}\) is homogeneous in the Sgrading.
Proof
Example 6.4
(Homogeneity with respect to the Sgrading)
The following example shows how redundant statements can seemingly complicate the situation and why it is advantageous to work with minimal sets of CIstatements defining a given CIideal. However, solving the conditional independence implication problem is difficult in general [14].
Example 6.5
The converse of Lemma 6.3 need not hold. Consider the ideal Open image in new window , which is {1}homogeneous. By Proposition 6.1 it equals the CIideal of Open image in new window which does not satisfy the combinatorial conditions in Lemma 6.3.
Lemma 6.6
\(J_{\mathcal{L}_{i}} = \langle \operatorname {Lift}(\mathcal{M}_{i}) \rangle\).
Proof
Definition 6.7
Theorem 6.8
Proof
Homogeneity in the (codimension zero) Sgrading follows from Lemma 6.3. The generators of the codimension zero toric fiber product on the right hand side consist of Lifts and Quads by [34] and, in the toric case, Sect. 4.1. Since the Quads correspond exactly to the independence statement \(\mathcal {S}\), the theorem is a consequence of Lemma 6.6. □
Example 6.9
6.2 Graphical conditional independence models
Our main motivation for toric fiber products together of CIideals comes from an application to the global Markov condition in graphical models. Let G be a simple undirected graph on the vertex set V.
Definition 6.10
Lemma 6.11
The global Markov ideal is a binomial ideal.
Proof
If a statement is valid on G but does not involve all vertices, then it is the consequence of a valid statement that does use all vertices. Indeed, if v∈V∖(A∪B∪C), then v cannot be connected to both A and B as then C would not separate. It is thus connected to at most one of them, say A. In this case Open image in new window is a valid statement for G. Now use the decomposition property, also valid for CIideals, Open image in new window to get the result. □
Assume that we can decompose the vertex set of G as V=V _{1}∪V _{2}, so that the induced subgraph on S:=V _{1}∩V _{2} is complete, and any path from V _{1} to V _{2} passes S. In this case S is a separator. Since a global Markov ideal is binomial it is Shomogeneous, and the same holds for the CIideals \(J_{G_{1}}\) and \(J_{G_{2}}\), arising from the induced subgraphs on V _{1} and V _{2}.
Theorem 6.12
Proof
As an immediate corollary we get the following known result [8, 37].
Corollary 6.13
The global Markov ideal of a chordal graph is prime.
Proof
A chordal graph decomposes as a product of its maximal cliques. Inductively applying Theorem 6.12 and the fact that the toric fiber product of geometrically prime ideals is geometrically prime, gives the result. □
The following corollary was one of our initial motivations for this section and Theorem 3.1.
Corollary 6.14
(Primary decompositions of graphical CIideals)
Let G be a graph with vertex set V=V _{1}∪V _{2} with V _{1}∩V _{2} a separator in G. Let G _{1} and G _{2} be the induced subgraphs on V _{1} and V _{2}, respectively. A primary decomposition of J _{ G } can be obtained from toric fiber products of the primary components of \(J_{G_{1}}\) and \(J_{G_{2}}\).
As the primary decompositions of the CIideals J _{ G } are unknown for most graphs, we do not know in which situations we can guarantee that the toric fiber products of irredundant primary decompositions of CIideals yield an irredundant primary decomposition. In concrete situations Corollary 3.3 and Lemma 3.4 can be used. For instance the primary decomposition of the chain of squares in Example 1.3 is irredundant. Explicit computation shows that none of the eight monomial minimal primes contains all monomials of a given multidegree, and the same holds, of course, for the toric ideal. By Corollary 3.3 the toric fiber products of the prime components yield an irredundant prime decomposition of the ideal of two squares glued along an edge. When gluing the next square the grading is different, but Lemma 3.4 guarantees that the hypothesis of Corollary 3.3 is still fulfilled. Unfortunately this argument cannot be applied to all conditional independence models, as the following example demonstrates.
Example 6.15
Notes
Acknowledgements
Alexander Engström gratefully acknowledges support from the Miller Institute for Basic Research in Science at UC Berkeley. Thomas Kahle was supported by an EPDI Fellowship. Seth Sullivant was partially supported by the David and Lucille Packard Foundation and the US National Science Foundation (DMS 0954865).
The authors are happy to thank the MittagLeffler institute for hosting them for the final part of this project, during the program on “Algebraic Geometry with a View towards Applications”. Johannes Rauh made valuable comments on an earlier version of the manuscript.
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