# On the strongly robustness property of toric ideals

Dimitra Kosta  ,  Apostolos Thoma  and  Marius Vladoiu  Dimitra Kosta, School
of Mathematics, University of Edinburgh and Maxwell Institute for Mathematical
Sciences, United Kingdom<EMAIL_ADDRESS>Apostolos Thoma, Department of
Mathematics, University of Ioannina, Ioannina 45110, Greece<EMAIL_ADDRESS>Marius Vladoiu, Faculty of Mathematics and Computer Science, University of
Bucharest, Str. Academiei 14, Bucharest, RO-010014, Romania, and  “Simion
Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764,
014700, Bucharest, Romania<EMAIL_ADDRESS>
###### Abstract.

To every toric ideal one can associate an oriented matroid structure,
consisting of a graph and another toric ideal, called bouquet ideal. The
connected components of this graph are called bouquets. Bouquets are of three
types; free, mixed and non mixed. We prove that the cardinality of the
following sets - the set of indispensable elements, minimal Markov bases, the
Universal Markov basis and the Universal Gröbner basis of a toric ideal -
depends only on the type of the bouquets and the bouquet ideal. These results
enable us to introduce the strongly robustness simplicial complex and show
that it determines the strongly robustness property. For codimension 2 toric
ideals, we study the strongly robustness simplicial complex and prove that
robustness implies strongly robustness.

###### Key words and phrases:

Toric ideals, Graver basis, Universal Gröbner basis, Markov bases, Robust
ideals

###### 2010 Mathematics Subject Classification:

13F65, 13P10, 14M25, 05C90, 62R01

Corresponding author: Dimitra Kosta

##  1\. Introduction

An ideal in a polynomial ring is called robust if it is minimally generated by
the Universal Gröbner basis (see [4]), where the Universal Gröbner basis is
the union of all reduced Gröbner bases. For toric ideals, the Universal
Gröbner basis is always contained in the Graver basis, see [22, Chapter 4]. If
$I_{A}$ is positively graded then every minimal system of generators of a
toric ideal $I_{A}$ is contained in the Graver basis, see [8, Theorem 2.3].
Thus there exists a stronger notion of robustness for positively graded toric
ideals. A positively graded toric ideal is called strongly robust if it is
minimally generated by its Graver basis, see [24]. It follows that for a
strongly robust toric ideal $I_{A}$ the following sets are identical: the set
of indispensable elements, any minimal system of generators, any reduced
Gröbner basis, the Universal Gröbner basis and the Graver basis, see for
example [19, Section 4]. Robust and strongly robust ideals have been studied
in several articles [2, 3, 4, 5, 10, 11, 13, 14, 19, 22, 23, 24, 26].

Strongly robust toric ideals appear in geometry, as defining ideals of non-
pyramidal self-dual projective toric varieties, see [6, 28]; in combinatorial
commutative algebra, as toric ideals of Lawrence matrices helping in the
computation of the Graver basis, see [22, Chapter 7]; in algebraic statistics,
as toric ideals whose Markov bases have the distance-reducing property, see
[24, 25]. Note that for a toric ideal being strongly robust implies being
robust. In [5], Boocher et al. wonder whether robust toric ideals are always
strongly robust. In fact, they proved this is the case for toric ideals of
graphs. On the other hand, based on [20, Theorem 4.2, Remark 4.4], using the
theory of bouquets, [19], it follows that if robust toric ideals of
hypergraphs are strongly robust then this is true for all positively graded
toric ideals.

Theorem 4.2 of [19] suggests a special relation between strongly robust ideals
and bouquets. For example, if all the bouquets are mixed then the toric ideal
is strongly robust, see [19, Corollary 4.4], thus Petrović et al. raised the
question whether all strongly robust toric ideals have at least one mixed
bouquet. This question was answered in affirmative by Sullivant [24] for
codimension 2 toric ideals based on the reduced Gale diagrams, a useful tool
given by Peeva and Sturmfels in [18] for studying problems related to
codimension 2 toric ideals. Note that reduced Gale diagrams have a strong
relationship with the bouquet structure and they also preserve information
about the mixedness or not of the bouquets, see Section 4 of this article.

In Section 2, we strengthen the bouquet results of [19] in the following
manner. To every bouquet we can assign a sign $+$ if it is non-mixed, $-$ if
it is mixed and $0$ if it is free. We prove that if two toric ideals have the
same bouquet ideal and the same signatures on the corresponding bouquets, then
there is a bijective correspondence between indispensables, Markov bases,
Gröbner bases and the universal Gröbner bases of the two ideals. The
corresponding result for circuits and the Graver bases was proved in [19]. In
Section 3, the previous results help us to define the strongly robustness
simplicial complex, about which we show that determines the strongly
robustness property for toric ideals, thus partially answering a question
posed by Boocher and Robeva [4, Question 3.5]. We show how one can use the
second Lawrence lifting to compute the strongly robustness simplicial complex.
Then the theory of generalized Lawrence matrices of [19, Section 2] produces
examples of strongly robust toric ideals; actually any strongly robust toric
ideal can be produced in this way. In Section 4, we use the results of [18,
24] in codimension 2 to provide a Hilbert basis method to compute the strongly
robustness simplicial complex and prove that, for codimension 2 toric ideals,
robustness implies strongly robustness. Finally, using [24, Lemma 2.7], we
prove an interesting property of codimension 2 ideals: if circuits are
indispensable then all elements of the Graver basis are indispensable.

##  2\. Bouquet Ideals

Let $K$ be a field and
$A=[{\mathbb{a}}_{1},\dots,{\mathbb{a}}_{n}]\in{\mathbb{Z}}^{m\times n}$ be an
integer matrix with the set of column vectors
$\\{{\mathbb{a}}_{1},\ldots,{\mathbb{a}}_{n}\\}$ such that
$\operatorname{Ker}_{{\mathbb{Z}}}(A)\cap{\mathbb{N}}^{n}=\\{{\mathbf{0}}\\}$.
The toric ideal of $A$ is the ideal $I_{A}\subset K[x_{1},\ldots,x_{n}]$
generated by the binomials ${x}^{{{\mathbb{u}}}^{+}}-{x}^{{{\mathbb{u}}}^{-}}$
where ${{\mathbb{u}}}\in\operatorname{Ker}_{{\mathbb{Z}}}(A),$ and ${\bf
u}={\bf u}^{+}-{\bf u}^{-}$ is the expression of ${\bf u}$ as a difference of
two non-negative vectors with disjoint support, see [22, Chapter 4]. The
condition $\operatorname{Ker}_{\mathbb{Z}}(A)\cap\mathbb{N}^{n}=\\{{\bf 0}\\}$
means that there is a strictly positive vector on the row span of the matrix
$A$, therefore the toric ideal is positively graded. To the matrix $A$ we
associate its Gale transform, which is the $n\times(n-r)$ matrix whose columns
span the lattice $\operatorname{Ker}_{{\mathbb{Z}}}(A)$, and $r$ is the rank
of $A$. We will denote the set of ordered row vectors of the Gale transform by
$\\{{\bf b}_{1},\dots,{\bf b}_{n}\\}$. The vector ${\bf a}_{i}$ is called free
if its Gale transform ${\bf b}_{i}$ is equal to the zero vector, which means
that $i$ is not contained in the support of any element in
$\operatorname{Ker}_{\mathbb{Z}}(A)$. The bouquet graph $G_{A}$ of $I_{A}$ is
the graph on the set of vertices $\\{{\bf a}_{1},\dots,{\bf a}_{n}\\}$, whose
edge set $E_{A}$ consists of those $\\{{\mathbb{a}}_{i},{\mathbb{a}}_{j}\\}$
for which ${\bf b}_{i}$ is a rational multiple of ${\bf b}_{j}$ and vice
versa. The connected components of the graph $G_{A}$ are called bouquets.

It follows from the definition that the free vectors of $A$ form one bouquet,
which we call the free bouquet of $G_{A}$. The non-free bouquets are of two
types: mixed and non-mixed. A non-free bouquet is mixed if contains an edge
$\\{{\mathbb{a}}_{i},{\mathbb{a}}_{j}\\}$ such that ${\bf b}_{i}=\lambda{\bf
b}_{j}$ for some $\lambda<0$, and is non-mixed if it is either an isolated
vertex or for all of its edges $\\{{\mathbb{a}}_{i},{\mathbb{a}}_{j}\\}$ we
have ${\bf b}_{i}=\lambda{\bf b}_{j}$ with $\lambda>0$, see [19, Lemma 1.2].

We reorder the vectors ${\mathbb{a}}_{1},\dots,{\mathbb{a}}_{n}$ to
${\mathbb{a}}_{11},{\mathbb{a}}_{12},\dots,{\mathbb{a}}_{1k_{1}},\
{\mathbb{a}}_{21},{\mathbb{a}}_{22},\dots,{\mathbb{a}}_{2k_{2}},\ \dots,\\\
{\mathbb{a}}_{s1},{\mathbb{a}}_{s2},\dots,{\mathbb{a}}_{sk_{s}}$ in such a way
the first $k_{1}$ vectors belong to the bouquet $B_{1}$, the next $k_{2}$ to
$B_{2}$ and so on up to the last $k_{s}$ that belong to the bouquet $B_{s}$.
Note that $k_{1}+k_{2}+\dots+k_{s}=n$. For each bouquet $B_{i}$ we define two
vectors ${\mathbb{c}}_{B_{i}}$ and ${\mathbb{a}}_{B_{i}}$ which record the
bouquet’s types and linear dependencies of the bouquet vectors. If the bouquet
$B_{i}$ is free then we set ${\mathbb{c}}_{B_{i}}\in{\mathbb{Z}}^{n}$ to be
any nonzero vector such that
$\operatorname{supp}({\mathbb{c}}_{B_{i}})=\\{i1,\dots,ik_{i}\\}$ and with the
property that the first nonzero coordinate, $c_{i1}$, is positive. For a non-
free bouquet $B_{i}$ of $A$, consider the Gale transforms of the elements in
$B_{i}$. All the Gale transforms are nonzero, since the bouquet is not free,
and pairwise linearly dependent, since they belong to the same bouquet.
Therefore, there exists a nonzero coordinate $l$ in all of them. Let
$g_{l}=\gcd(({\bf b}_{i1})_{l},({\bf b}_{i2})_{l}),\dots,({\bf
b}_{ik_{i}})_{l})$, where $({\bf w})_{l}$ is the $l$-th component of a vector
${\bf w}$. Then ${\bf c}_{B_{i}}$ is the vector in ${\mathbb{Z}}^{n}$ whose
$qj$-th component is $0$ if $q\not=i$, and $c_{ij}=\varepsilon_{i1}({\bf
b}_{ij})_{l}/g_{l}$, where $\varepsilon_{i1}$ represents the sign of the
integer $({\bf b}_{i1})_{l}$. Note that $c_{i1}$ is always positive. Then the
vector ${\mathbb{a}}_{B_{i}}$ (see [19, Definition 1.7]), is defined as
${\mathbb{a}}_{B_{i}}=\sum_{j=1}^{n}(c_{B_{i}})_{j}{\mathbb{a}}_{j}\in{\mathbb{Z}}^{m}$.

If $B_{i}$ is a non-free bouquet of $A$, then $B_{i}$ is a mixed bouquet if
and only if the vector ${\bf c}_{B_{i}}$ has a negative and a positive
coordinate, see [19, Lemma 1.6], and $B_{i}$ is non-mixed if and only if the
vector ${\mathbb{c}}_{B_{i}}$ has all nonzero coordinates positive. Let
$A_{B}$ be the matrix with columns the vectors ${\mathbb{a}}_{B_{i}}$, $1\leq
i\leq s$ then the toric ideal $I_{A_{B}}$ is the bouquet ideal of $A$.

Let
${\mathbf{u}}=(u_{1},u_{2},\ldots,u_{s})\in\operatorname{Ker}_{{\mathbb{Z}}}({A_{B}})$
then the linear map

$D({\mathbf{u}})=(c_{11}u_{1},c_{12}u_{1},\ldots,c_{1k_{1}}u_{1},c_{21}u_{2},\ldots,c_{2k_{2}}u_{2},\ldots,c_{s1}u_{s},\ldots,c_{sk_{s}}u_{s}),$

where all $c_{j1},1\leq j\leq s$, are positive, is an isomorphism from
$\operatorname{Ker}_{{\mathbb{Z}}}({A_{B}})$ to
$\operatorname{Ker}_{{\mathbb{Z}}}(A)$, see [19, Theorem 1.9]. Therefore,
since the matrices $A$ and $A_{B}$ have their kernels isomorphic, their toric
ideals $I_{A}$ and $I_{A_{B}}$ have the same codimension.

Note that the bouquet ideal is simple: a toric ideal is called simple if every
bouquet is a singleton, in other words if $I_{T}\subset K[x_{1},\dots,x_{s}]$
and has $s$ bouquets. The bouquet ideal of a simple toric ideal $I_{A}$ is
$I_{A}$. All toric ideals $I_{A}\subset K[x_{1},\dots,x_{n}]$ of height one,
the ideals of monomial curves, are simple if $n\geq 3$. There are no simple
toric ideals in codimension one, except the zero ideal in the polynomial ring
in one variable.

######  Definition 2.1.

Let $I_{T}\subset K[x_{1},\dots,x_{s}]$ be a simple toric ideal and
$\omega\subset\\{1,\dots,s\\}$. A toric ideal $I_{A}$ is called
$T_{\omega}$-robust ideal if and only if

  * •

the bouquet ideal of $I_{A}$ is $I_{T}$ and

  * •

$\omega=\\{i\in[s]|B_{i}\ \text{is non mixed}\\}.$

The $T_{\emptyset}$-robust toric ideals are always strongly robust ideals, see
[19, Section 4], while the $T_{[s]}$-robust toric ideals are the stable
ideals, see [19, Section 3]. Every toric ideal $I_{A}$ is a
$T_{\omega}$-robust ideal where $I_{T}$ is the bouquet ideal of $I_{A}$ and
$\omega$ is the set of indices corresponding to the non mixed bouquets of
$I_{A}$. The complement of $\omega$ in $[s]$ is the set of indices
corresponding to the free and the mixed bouquets. The ideal $I_{A}$ has a free
bouquet if and only if the bouquet ideal $I_{T}$ has a free bouquet,
consisting of one free vector since $I_{T}$ is simple. Thus the combination of
the information given by $\omega$ and $T$ provides the signatures of all the
bouquets of a $T_{\omega}$-robust toric ideal. The results of this section can
be summarized: the cardinality of the sets of different toric bases depend
only on the signatures of the bouquets and the bouquet ideal. Remark that in
[19, Theorem 1.11] it was proved that the cardinality of the set of circuits
and of the Graver basis, respectively, depends only on the bouquet ideal. Also
in the case of any two $T_{[s]}$-robust toric ideals there is a one-to-one
correspondence between all different toric bases, [19, Theorem 3.7]. The aim
of this section is to generalize these results for all toric ideals with the
same signatures and the same bouquet ideal, by proving that in these cases
there exists also a one-to-one correspondence between the indispensable
elements, minimal Markov bases, the universal Markov bases (see Theorem 2.2),
reduced Gröbner bases and the universal Gröbner bases (see Theorem 2.3).

A Markov basis of $A$ is a finite subset $\mathcal{M}$ of
$\operatorname{Ker}_{\mathbb{Z}}(A)$ such that whenever
$\textbf{w},\textbf{v}\in\mathbb{N}^{n}$ and
$\textbf{w}-\textbf{v}\in\operatorname{Ker}_{\mathbb{Z}}(A)$ there exists a
subset $\\{\textbf{u}_{i}:i=1,\ldots,r\\}$ of $\mathcal{M}$ that connects w to
v. Here, connectedness means that
$\textbf{w}-\textbf{v}=\sum_{i=1}^{r}\textbf{u}_{i}$, and
$(\textbf{w}-\sum_{i=1}^{p}\textbf{u}_{i})\in\mathbb{N}^{n}$ for all $1\leq
p\leq r$ and $\textbf{u}_{i}\in\mathcal{M}$ means that
$\textbf{u}_{i}\in\mathcal{M}$ or the opposite
$-\textbf{u}_{i}\in\mathcal{M}$. (We will also express that
$\textbf{w}\in\mathbb{N}^{n}$ by writing $\textbf{w}\geq\textbf{0}$, that is,
each component $w_{i}$ of ${\mathbb{w}}$ is non-negative.) A Markov basis
$\mathcal{M}$ is minimal if no proper subset of $\mathcal{M}$ is a Markov
basis. The universal Markov basis $\mathcal{M}(A)$ of $A$ is the union of all
minimal Markov bases of $A$, up to identification of opposite vectors, and the
set of indispensable elements $S(A)$ is the intersection of all different
minimal Markov bases, via the same identification. By a classical result of
Diaconis and Sturmfels [12] minimal Markov bases of $A$ correspond to minimal
system of generators for the toric ideal $I_{A}$: $\mathcal{M}$ is minimal
Markov basis of $A$ if and only if
$\\{\textbf{x}^{u^{+}}-\textbf{x}^{u^{-}}|\textbf{u}\in\mathcal{M}\\}$ is a
minimal system of binomial generators for the toric ideal $I_{A}$. The set of
indispensable binomials $S(I_{A})$ of the toric ideal is the set
$\\{\textbf{x}^{u^{+}}-\textbf{x}^{u^{-}}|\textbf{u}\in S(A)\\}$, that is the
intersection of all minimal systems of binomial generators of $I_{A}$, up to
identification of opposite binomials.

For the proofs of the next two theorems we need the following notation. Let
$B_{1}^{(j)},\ldots,B_{s}^{(j)}$ be the bouquets of the toric ideal
$I_{A_{j}}$ with
${\mathbb{c}}_{B_{1}^{(j)}},\ldots,{\mathbb{c}}_{B_{s}^{(j)}}$ the
corresponding vectors for $j=1,2$. Furthermore, we denote by $k_{i}$ and
$l_{i}$, respectively, the number of elements of the $i$-th bouquet of
$B_{i}^{(1)}$ and $B_{i}^{(2)}$, respectively, and by
$c^{(1)}_{i1},\ldots,c^{(1)}_{ik_{i}}$ and
$c^{(2)}_{i1},\ldots,c^{(2)}_{il_{i}}$ the nonzero coordinates of
${\mathbb{c}}_{B_{i}^{(1)}}$ and ${\mathbb{c}}_{B_{i}^{(2)}}$, respectively.
Furthermore, associated to the matrices $A_{1}$ and $A_{2}$ we consider the
toric ideals $I_{A_{1}}\subset
K[x_{11},\ldots,x_{1k_{1}},\ldots,x_{s1},\ldots,x_{sk_{s}}]$ and
$I_{A_{2}}\subset
K[y_{11},\ldots,y_{1l_{1}},\ldots,y_{s1},\ldots,y_{sl_{s}}]$. According to
[19, Theorem 1.9], for each $j=1,2$ there is a one-to-one correspondence
between the elements of $\operatorname{Ker}_{\mathbb{Z}}(T)$ and the elements
of $\operatorname{Ker}_{\mathbb{Z}}(A_{j})$ given by the map
$D_{j}:\textbf{u}\mapsto D_{j}(\textbf{u})$, which sends the element
$\textbf{u}=(u_{1},\ldots,u_{s})\in\operatorname{Ker}_{\mathbb{Z}}(T)$ to
$D_{j}(\textbf{u}):={\mathbb{c}}_{B_{1}^{(j)}}u_{1}+\cdots+{\mathbb{c}}_{B_{s}^{(j)}}u_{s}\in\operatorname{Ker}_{{\mathbb{Z}}}(A_{j})$.
Therefore, all elements of $\operatorname{Ker}_{\mathbb{Z}}(A_{j})$ are in the
form $D_{j}(\textbf{u})$ for a unique
$\textbf{u}\in\operatorname{Ker}_{\mathbb{Z}}(T)$.

######  Theorem 2.2.

Let $I_{A_{1}},I_{A_{2}}$ be two $T_{\omega}$-robust toric ideals. If
$\mathcal{M}_{1}$ is a minimal Markov basis of $A_{1}$ then
$\mathcal{M}_{2}=\\{D_{2}(\textbf{u})|\
{\mathbb{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)\text{ such that
}D_{1}(\textbf{u})\in\mathcal{M}_{1}\\}$ is a minimal Markov basis of $A_{2}$.
Thus there exists a one-to-one correspondence between the sets of all
different minimal Markov bases of the two ideals. In particular, there exists
a one-to-one correspondence between the indispensable elements and the
universal Markov bases of ${A_{1}}$ and ${A_{2}}$, respectively.

###### Proof.

Let $\mathcal{M}_{1}$ be a minimal Markov basis of $A_{1}$. First we want to
prove that $\mathcal{M}_{2}=\\{D_{2}(\textbf{u})|\
{\mathbb{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)\text{ such that
}D_{1}(\textbf{u})\in\mathcal{M}_{1}\\}$ is a Markov basis of $A_{2}$. It is
enough to prove that one can connect $D_{2}(\textbf{v})^{+}$ to
$D_{2}(\textbf{v})^{-}$ for every
$\textbf{v}\in\operatorname{Ker}_{\mathbb{Z}}(T)$.

Since $\mathcal{M}_{1}$ is a Markov basis of $A_{1}$ there exists a subset
$\\{D_{1}(\textbf{u}_{j})|\ j=1,\ldots r\\}$ of $\mathcal{M}_{1}$ such that
for the element $D_{1}(\textbf{v})\in\operatorname{Ker}_{\mathbb{Z}}(A_{1})$
we have
$D_{1}(\textbf{v})^{+}-D_{1}(\textbf{v})^{-}=\sum_{j=1}^{r}D_{1}(\bf{u}_{j})$
and $D_{1}(\textbf{v})^{+}-\sum_{j=1}^{p}D_{1}(\textbf{u}_{j})\geq\textbf{0}$
for all $1\leq p\leq r$. Therefore, we have that

$\left({\mathbb{c}}_{B_{i}^{(1)}}v_{i}\right)^{+}-\left({\mathbb{c}}_{B_{i}^{(1)}}u_{1i}+\cdots+{\mathbb{c}}_{B_{i}^{(1)}}u_{pi}\right)\geq\textbf{0}\
\ \ \ \forall\ 1\leq p\leq r,\ \forall\ 1\leq i\leq s,$

which we can further write at coordinate level as

$\left(\left(c_{i1}^{(1)},c_{i2}^{(1)},\ldots,c_{ik_{i}}^{(1)}\right)v_{i}\right)^{+}\geq\left(c_{i1}^{(1)},c_{i2}^{(1)},\ldots,c_{ik_{i}}^{(1)}\right)u_{1i}+\cdots+\left(c_{i1}^{(1)},c_{i2}^{(1)},\ldots,c_{ik_{i}}^{(1)}\right)u_{pi},$

for all $1\leq p\leq r$ and for all $1\leq i\leq s$.

Case 1: Let $i\in\omega,v_{i}>0$. In this case the bouquet $B_{i}^{(1)}$ is
non-mixed and all $c_{ij}^{(1)}>0$. We have for all $1\leq p\leq r$ and for
all $1\leq j\leq k_{i}$

$c^{(1)}_{ij}v_{i}\geq c^{(1)}_{ij}\left(u_{1i}+\cdots+u_{pi}\right)$

which gives

$v_{i}\geq u_{1i}+\cdots+u_{pi}.$

Since the toric ideal $I_{A_{2}}$ is $T_{\omega}$-robust, $B_{i}^{(2)}$ is
also non-mixed and all $c_{ij}^{(2)}>0$. Therefore, the last set of
inequalities becomes by multiplication with $c_{ij}^{(2)}$

$c^{(2)}_{ij}v_{i}\geq c^{(2)}_{ij}(u_{1i}+\cdots+u_{pi})\ \ \ \ \ \forall\
1\leq j\leq l_{i},$

which implies

$\left(\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)v_{i}\right)^{+}\geq\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{1i}+\cdots+\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{pi},$

for all $1\leq p\leq r$.

Case 2: Let $i\in\omega,v_{i}\leq 0$. Here again the bouquet $B_{i}^{(1)}$ is
non-mixed and $c_{ij}^{(1)}>0$ for all $j=1,2,\ldots,k_{i}$. We have

$0\geq c^{(1)}_{ij}(u_{1i}+\cdots+u_{pi})$

which gives

$0\geq u_{1i}+\cdots+u_{pi}.$

Again the toric ideal $I_{A_{2}}$ is $T_{\omega}$-robust, $B_{i}^{(2)}$ is
also non-mixed and all $c_{ij}^{(2)}>0$, $j=1,2,\ldots,l_{i}$. Therefore, the
last set of inequalities becomes by multiplication with $c_{ij}^{(2)}$

$0\geq c_{ij}^{(2)}(u_{1i}+\cdots+u_{pi})\ \ \ \ \forall\ 1\leq j\leq l_{i},$

which implies

$\left(\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)v_{i}\right)^{+}\geq\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{1i}+\cdots+\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{pi},$

for all $1\leq p\leq r$.

Case 3: Let $i\notin\omega$ and the bouquet $B_{i}^{(1)}$ is the free bouquet
of $I_{A_{1}}$, if it has a free bouquet. Since
$\textbf{v}\in\operatorname{Ker}_{\mathbb{Z}}(T)$ and $B_{i}^{(1)}$ is the
free bouquet then $v_{i}$ is zero. Since this is true for all elements of
$\operatorname{Ker}_{\mathbb{Z}}(T)$ it means $I_{A_{2}}$ has a free bouquet,
the $B_{i}^{(2)}$, and all $v_{i}=u_{1i}=\dots=u_{pi}=0.$ Thus

$\left(\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)v_{i}\right)^{+}=\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{1i}+\cdots+\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{pi}={\bf
0}.$

Case 4: Let $i\notin\omega,v_{i}>0$ and the bouquet $B_{i}^{(1)}$ is mixed,
namely the vector ${\mathbb{c}}_{B_{i}^{(1)}}$ has both positive and negative
coordinates and splits as
${\mathbb{c}}_{B_{i}^{(1)}}=\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{+}-\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{-}$.
We have

$\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{+}v_{i}\geq\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{+}\left(u_{1i}+\cdots+u_{pi}\right)$

and

${\bf 0}\geq-({\mathbb{c}}_{B_{i}^{(1)}})^{-}(u_{1i}+\cdots+u_{pi}).$

Combined together they give $v_{i}\geq u_{1i}+\cdots+u_{pi}\geq 0.$ Since the
toric ideals $I_{A_{1}}$ and $I_{A_{2}}$ are $T_{\omega}$-robust, whenever
$B_{i}^{(1)}$ is mixed, $B_{i}^{(2)}$ is also mixed. In this case
${\mathbb{c}}_{B_{i}^{(2)}}$ has both positive and negative coordinates and
splits as
${\mathbb{c}}_{B_{i}^{(2)}}=\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{+}-\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{-}$.
We have

$\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{+}v_{i}\geq\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{+}(u_{1i}+\cdots+u_{pi})$

and

${\bf
0}\geq-\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{-}(u_{1i}+\cdots+u_{pi})$

which imply

$\left(\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)v_{i}\right)^{+}\geq\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{1i}+\cdots+\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{pi},$

for all $1\leq p\leq r$.

Case 5: Let $i\notin\omega,v_{i}\leq 0$ and the bouquet $B_{i}^{(1)}$ is
mixed, namely the vector ${\mathbb{c}}_{B_{i}^{(1)}}$ has both positive and
negative coordinates and splits as
${\mathbb{c}}_{B_{i}^{(1)}}=\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{+}-\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{-}$.
We have

$-\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{-}v_{i}\geq-\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{-}(u_{1i}+\cdots+u_{pi})$

and

$\textbf{0}\geq\left({\mathbb{c}}_{B_{i}^{(1)}}\right)^{+}(u_{1i}+\cdots+u_{pi}).$

Combined together they give $v_{i}\leq u_{1i}+\cdots+u_{pi}\leq 0.$ By
multiplying appropriately the above inequalities by either
$\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{+}$ or
$\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{-}$ we have

$-\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{-}v_{i}\geq-\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{-}(u_{1i}+\cdots+u_{pi})$

and

$\textbf{0}\geq\left({\mathbb{c}}_{B_{i}^{(2)}}\right)^{+}(u_{1i}+\cdots+u_{pi})$

which imply

$\left(\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)v_{i}\right)^{+}\geq\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{1i}+\cdots+\left(c_{i1}^{(2)},c_{i2}^{(2)},\ldots,c_{il_{i}}^{(2)}\right)u_{pi},$

for all $1\leq p\leq r$.

Finally combining all five cases we arrive at

$\left(D_{2}(\textbf{v})^{+}-\sum_{j=1}^{p}D_{2}(\textbf{u}_{j})\right)\geq
0\text{ for all }1\leq p\leq r.$

Also, we have that
$D_{1}(\textbf{v})^{+}-D_{1}(\textbf{v})^{-}=D_{1}(\textbf{v})=\sum_{j=1}^{r}D_{1}(\textbf{u}_{j})\Rightarrow
D_{2}(\textbf{v})^{+}-D_{2}(\textbf{v})^{-}=D_{2}(\textbf{v})=\sum_{j=1}^{r}D_{2}(\bf{u}_{j}),$
since both $D_{1},D_{2}$ are linear maps.

Therefore, this concludes that for any element
$D_{2}(\textbf{v})\in\operatorname{Ker}_{\mathbb{Z}}(A_{2})$, there exists a
subset $\\{D_{2}(\textbf{u}_{j})|\ j=1,\ldots r\\}$ of $\mathcal{M}_{2}$ that
connects $D_{2}(\textbf{v})^{+}$ and $D_{2}(\textbf{v})^{-}$.

This one-to-one correspondence between Markov bases of $A_{1}$ and $A_{2}$,
ensures also that the map $f:D_{1}(\textbf{u})\mapsto D_{2}(\bf{u})$ preserves
the minimality of the Markov bases. Indeed, if $\mathcal{M}_{1}$ is a minimal
Markov basis of $A_{1}$ then, as we saw,
$\mathcal{M}_{2}=\\{D_{2}(\textbf{u})|\
{\mathbb{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)\text{ such that
}D_{1}(\bf{u})\in\mathcal{M}_{1}\\}$ is a Markov basis of $A_{2}$, and if it
were not minimal then a proper subset $\mathcal{M}_{2}^{{}^{\prime}}$ of
$\mathcal{M}_{2}$ would be a minimal Markov basis, and thus by repeating the
previous argument we would obtain that a proper subset
$\mathcal{M}_{1}^{{}^{\prime}}=\\{D_{1}({\bf u})|\
{\mathbb{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)\text{ such that
}D_{2}({\bf u})\in\mathcal{M}_{2}^{{}^{\prime}}\\}$ of the minimal Markov
basis $\mathcal{M}_{1}$ would be a Markov basis of $A_{1}$, a contradiction.

The set of indispensable elements is the intersection of all different minimal
Markov bases and the universal Markov basis is the union of all minimal Markov
bases. Therefore it follows that $f:D_{1}(\textbf{u})\mapsto D_{2}(\bf{u})$ is
a one-to-one correspondence between the two sets of the indispensable elements
and the same is true for the universal Markov bases of $A_{1}$ and $A_{2}$.

######  Theorem 2.3.

Let $I_{A_{1}},I_{A_{2}}$ be two $T_{\omega}$-robust toric ideals. If
$\mathcal{G}$ is a reduced Gröbner basis of $I_{A_{1}}$ with respect to a
monomial order $<_{1}$ then there exists a monomial order $<_{2}$ such that
$\mathcal{G}^{\prime}=\\{y^{D_{2}(\textbf{u})^{+}}-y^{D_{2}(\textbf{u})^{-}}|\
{\mathbb{u}}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)\text{ s.t. }\
x^{D_{1}(\textbf{u})^{+}}-x^{D_{1}(\textbf{u})^{-}}\in\mathcal{G}\\}$ is a
reduced Gröbner basis of $I_{A_{2}}$. Furthermore, if

$\operatorname{in}_{<_{1}}(I_{A_{1}})=(x^{D_{1}({\bf
u_{1}})^{+}},\ldots,x^{D_{1}({\bf u_{r}})^{+}})$

then

$\operatorname{in}_{<_{2}}(I_{A_{2}})=(y^{D_{2}({\bf
u_{1}})^{+}},\ldots,y^{D_{2}({\bf u_{r}})^{+}}),$

for some ${\bf u}_{1},\ldots,{\bf
u}_{r}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)$. Thus there exists a one-to-one
correspondence between the reduced Gröbner bases of the two ideals. In
particular, there is a one-to-one correspondence between the universal Gröbner
bases of $I_{A_{1}}$ and $I_{A_{2}}$.

###### Proof.

It is enough to prove that if $\mathcal{G}=\\{x^{D_{1}({\bf
u}_{1})^{+}}-x^{D_{1}({\bf u}_{1})^{-}},\ldots,x^{D_{1}({\bf
u}_{r})^{+}}-x^{D_{1}({\bf u}_{r})^{-}}\\}$ is a reduced Gröbner basis of the
toric ideal $I_{A_{1}}$ with respect to the monomial order $<_{1}$ on
$K[x_{11},\ldots,x_{1{k_{1}}},\ldots,x_{s1},\ldots,x_{s{k_{s}}}]$ for some
${\bf u}_{1},\ldots,{\bf u}_{r}\in\operatorname{Ker}_{\mathbb{Z}}(T)$, then
there exists a monomial order $<_{2}$ on
$K[y_{11},\ldots,y_{1l_{1}},\ldots,y_{s1},\ldots,y_{sl_{s}}]$ with respect to
which

$\mathcal{G}^{\prime}=\left\\{y^{D_{2}({\bf u}_{1})^{+}}-y^{D_{2}({\bf
u}_{1})^{-}},\ldots,y^{D_{2}({\bf u}_{r})^{+}}-y^{D_{2}({\bf
u}_{r})^{-}}\right\\}$

is a reduced Gröbner basis of the toric ideal $I_{A_{2}}$.

We recall that the nonzero coordinates of ${\mathbb{c}}_{B_{i}^{(1)}}$ are
$c_{i1}^{(1)},c_{i2}^{(1)},\ldots,c_{ik_{i}}^{(1)}$. We write each vector
${\mathbb{c}}_{B_{i}^{(1)}}$ as
${\mathbb{c}}_{B_{i}^{(1)}}^{+}-{\mathbb{c}}_{B_{i}^{(1)}}^{-}$, where for
$i\in\omega$ we have ${\mathbb{c}}_{B_{i}^{(1)}}^{-}=\mathbf{0}$. The
coordinates of ${\mathbb{c}}_{B_{i}^{(1)}}^{+}$ and
${\mathbb{c}}_{B_{i}^{(1)}}^{-}$ are denoted by $c_{ij}^{(1)+}$ and
$c_{ij}^{(1)-}$, respectively. One of them is zero and the other
$|c_{ij}^{(1)}|$. Similarly, one defines the corresponding notation for the
coordinates of ${\mathbb{c}}_{B_{i}^{(2)}}$.

By [22, Proposition 1.11], there exists a weight vector
${\mathbb{w}}\in{\mathbb{N}}^{k_{1}+\cdots+k_{s}}$, say
${\mathbb{w}}=(w_{11},\ldots,w_{1k_{1}},\ldots,w_{s1},\ldots,w_{sk_{s}})$ such
that
$\operatorname{in}_{<_{1}}(I_{A_{1}})=\operatorname{in}_{\mathbf{w}}(I_{A_{1}})$.
Without loss of generality we may assume that $x^{D_{1}({\bf
u}_{p})^{+}}>_{1}x^{D_{1}({\bf u}_{p})^{-}}$, i.e. the dot product
$\mathbf{w}\cdot D_{1}({\bf u}_{p})>0$, for any $p=1,\ldots,r$. Thus

$\dots+\left(c_{i1}^{(1)+}w_{i1}+\dots+c_{i{k_{i}}}^{(1)+}w_{i{k_{i}}}\right)u_{pi}-\left(c_{i1}^{(1)-}w_{i1}+\dots+c_{i{k_{i}}}^{(1)-}w_{i{k_{i}}}\right)u_{pi}+\dots>0.$

We denote by
${\mathbb{c}}_{B_{i}^{(1)}}^{+}\cdot\mathbf{w}=c_{i1}^{(1)+}w_{i1}+c_{i2}^{(1)+}w_{i2}+\dots+c_{ik_{i}}^{(1)+}w_{ik_{i}}$
and by
${\mathbb{c}}_{B_{i}^{(1)}}^{-}\cdot\mathbf{w}=c_{i1}^{(1)-}w_{i1}+c_{i2}^{(1)-}w_{i2}+\dots+c_{ik_{i}}^{(1)-}w_{ik_{i}}$.
Next, if we let $r^{+}_{i}$ be the number of positive elements in
${\mathbb{c}}_{B_{i}^{(2)}}$ and $r^{-}_{i}$ the number of negative elements
in ${\mathbb{c}}_{B_{i}^{(2)}}$, then we define the vector
$\mathbf{w}^{\prime}\in\mathbb{N}^{l_{1}+\cdots+l_{s}}$ such that its $it$-th
coordinate is equal to
$\frac{{\mathbb{c}}_{B_{i}^{(1)}}^{+}\cdot\mathbf{w}}{c^{(2)}_{it}r^{+}_{i}}$
if $c^{(2)}_{it}>0$ and
$\frac{{\mathbb{c}}_{B_{i}^{(1)}}^{-}\cdot\mathbf{w}}{-c^{(2)}_{it}r^{-}_{i}}$
if $c^{(2)}_{it}<0$. Note that always ${r_{i}}^{+}\geq 1$, but ${r_{i}}^{-}=0$
for non mixed bouquets. In this last case there is no reason to define the
number
$\frac{{\mathbb{c}}_{B_{i}^{(1)}}^{-}\cdot\mathbf{w}}{-c^{(2)}_{it}r^{-}_{i}}$
since $c^{(2)}_{it}$ is always greater than $0$.

Let $\prec$ be an arbitrary monomial order on
$K[y_{11},\ldots,y_{1l_{1}},\ldots,y_{s1},\ldots,y_{sl_{s}}]$. We define by
$<_{2}$ the monomial order $\prec_{\mathbf{w}^{\prime}}$ on
$K[y_{11},\ldots,y_{1l_{1}},\ldots,y_{s1},\ldots,y_{sl_{s}}]$ induced by
$\prec$ and $\mathbf{w}^{\prime}$, see [22, Chapter 1] for definition. Next we
prove that $x^{D_{1}({\bf u})^{+}}>_{1}x^{D_{1}({\bf u})^{-}}$ implies that
$y^{D_{2}({\bf u})^{+}}>_{2}y^{D_{2}({\bf u})^{-}}$. Indeed, since
$x^{D_{1}({\bf u})^{+}}>_{1}x^{D_{1}({\bf u})^{-}}$ then $\mathbf{w}\cdot
D_{1}({\bf u})>0$, where ${\bf u}=(u_{1},\ldots,u_{s})$. It follows now from
the definition of $\mathbf{w}^{\prime}$ that

$\mathbf{w}^{\prime}\cdot D_{2}({\bf u})=\mathbf{w}\cdot D_{1}({\bf u})>0,$

which implies $y^{D_{2}({\bf u})^{+}}>_{2}y^{D_{2}({\bf u})^{-}}$, as desired.
In particular, we obtain that $y^{D_{2}({\bf u_{p}})^{+}}>_{2}y^{D_{2}({\bf
u_{p}})^{-}}$ for all $p=1,\ldots,r$, and as a consequence

$\left(y^{D_{2}({\bf u_{1}})^{+}},\ldots,y^{D_{2}({\bf
u_{r}})^{+}}\right)\subset\operatorname{in}_{<_{2}}(I_{A_{2}}).$

For the converse inclusion, let $y^{D_{2}({\bf u})^{+}}-y^{D_{2}({\bf
u})^{-}}\in I_{A_{2}}$ be an arbitrary element, and say that $y^{D_{2}({\bf
u})^{+}}>_{2}y^{D_{2}({\bf u})^{-}}$, the other case being similar. It follows
from the previous considerations that $x^{D_{1}({\bf
u})^{+}}>_{1}x^{D_{1}({\bf u})^{-}}$ and thus there exists an integer $p$ such
that $x^{D_{1}({\bf u_{p}})^{+}}|x^{D_{1}({\bf u})^{+}}$. That means
$x_{ij}^{(D_{1}({\bf u_{p}})^{+})_{ij}}\big{|}x_{ij}^{(D_{1}({\bf
u})^{+})_{ij}}$ for every $i$, $j$, where $(D_{1}({\bf u_{p}})^{+})_{ij}$ and
$(D_{1}({\bf u})^{+})_{ij}$ represent the $ij$-th coordinate of the vectors
$D_{1}({\bf u_{p}})^{+}$ and $D_{1}({\bf u})^{+}$, respectively. The exponent
of $x_{ij}$ in $x^{D_{1}({\bf u_{p}})^{+}}$ is
$c_{ij}^{(1)+}u_{pi}^{+}+c_{ij}^{(1)-}u_{pi}^{-}$, while in $x^{D_{1}({\bf
u})^{+}}$ is $c_{ij}^{(1)+}u_{i}^{+}+c_{ij}^{(1)-}u_{i}^{-}$. We consider the
following cases:
Case 1: Let $i\in\omega,u_{i}>0$. In this case the bouquets $B_{i}^{(1)}$,
$B_{i}^{(2)}$ are non-mixed and all $c_{ij}^{(1)}>0$, $c_{it}^{(2)}>0$. Then
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+c_{ij}^{(1)-}u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}u_{i}^{+}+c_{ij}^{-}u_{i}^{-}}$
means
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+0u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}u_{i}^{+}+0u_{i}^{-}}$.
Thus $u_{pi}^{+}\leq u_{i}$. Therefore
$y_{it}^{c_{it}^{(2)+}u_{pi}^{+}+0u_{pi}^{-}}\Big{|}y_{it}^{c_{it}^{(2)+}u_{i}^{+}+0}$
and thus $y_{it}^{(D_{2}({\bf u_{p}})^{+})_{it}}\big{|}y_{it}^{(D_{2}({\bf
u})^{+})_{it}}$.
Case 2: Let $i\in\omega,u_{i}\leq 0$. In this case the bouquets $B_{i}^{(1)}$,
$B_{i}^{(2)}$ are non-mixed and all $c_{ij}^{(1)}>0$, $c_{it}^{(2)}>0$. Then
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+c_{ij}^{(1)-}u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}u_{i}^{+}+c_{ij}^{(1)-}u_{i}^{-}}$
implies that $x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+0u_{pi}^{-}}$ divides
$x_{ij}^{c_{ij}^{(1)+}0+0u_{i}^{-}}=1$, and thus $u_{pi}^{+}=0$. Therefore
$1=y_{it}^{c_{it}^{(2)+}0+0u_{pi}^{-}}\Big{|}y_{it}^{c_{it}^{(2)+}u_{i}^{+}+0u_{i}^{-}}=1$
and consequently $y_{it}^{(D_{2}({\bf
u_{p}})^{+})_{it}}\big{|}y_{it}^{(D_{2}({\bf u})^{+})_{it}}$.
Case 3: Let $i\not\in\omega$, and $B^{(1)}_{i},B^{(2)}_{i}$ are free. In this
case, since $\bf{u_{p}},\bf{u}\in\operatorname{Ker}_{{\mathbb{Z}}}(T)$, then
both $u_{pi},u_{i}$ are zero and consequently
$1=y_{it}^{c_{it}^{(2)+}u_{pi}^{+}+c_{it}^{(2)-}u_{pi}^{-}}\Big{|}y_{it}^{c_{it}^{(2)+}u_{i}^{+}+c_{it}^{(2)-}u_{i}^{-}}=1$.
Therefore, we obtain $y_{it}^{(D_{2}({\bf
u_{p}})^{+})_{it}}\big{|}y_{it}^{(D_{2}({\bf u})^{+})_{it}}$.
Case 4: Let $i\not\in\omega$, $u_{i}>0$, and $B^{(1)}_{i},B^{(2)}_{i}$ are
mixed. In this case there is at least one $c_{ij}^{(1)}>0$ and at least one
$c_{ik}^{(1)}<0$. For $ij$ such that $c_{ij}^{(1)}>0$ the relation
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+c_{ij}^{(1)-}u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}u_{i}^{+}+c_{ij}^{(1)-}u_{i}^{-}}$
implies
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+0u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}u_{i}^{+}+0u_{i}^{-}}$.
Thus we have $u_{pi}^{+}\leq u_{i}$. For $ik$ such that $c_{ik}^{(1)}<0$ the
relation
$x_{ik}^{c_{ik}^{(1)+}u_{pi}^{+}+c_{ik}^{(1)-}u_{pi}^{-}}\Big{|}x_{ik}^{c_{ik}^{(1)+}u_{i}^{+}+c_{ik}^{(1)-}u_{i}^{-}}$
implies
$x_{ik}^{0u_{pi}^{+}+c_{ik}^{(1)-}u_{pi}^{-}}\Big{|}x_{ik}^{0u_{i}^{+}+c_{ik}^{(1)-}0}=1$,
and then $u_{pi}^{-}=0$. Summing up, we have obtained $0\leq
u_{pi}^{+}=u_{pi}\leq u_{i}$. Therefore
$y_{it}^{c_{it}^{(2)+}u_{pi}^{+}+c_{it}^{(2)-}u_{pi}^{-}}\Big{|}y_{it}^{c_{it}^{(2)+}u_{i}^{+}+c_{it}^{(2)-}u_{i}^{-}}$,
and thus $y_{it}^{(D_{2}({\bf u_{p}})^{+})_{it}}\big{|}y_{it}^{(D_{2}({\bf
u})^{+})_{it}}$.
Case 5: Let $i\not\in\omega$, $u_{i}\leq 0$, and $B^{(1)}_{i},B^{(2)}_{i}$ are
mixed. In this case there is at least one $c_{ij}^{(1)}>0$ and at least one
$c_{ik}^{(1)}<0$. For $ij$ such that $c_{ij}^{(1)}>0$ the relation
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+c_{ij}^{(1)-}u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}u_{i}^{+}+c_{ij}^{(1)-}u_{i}^{-}}$
implies
$x_{ij}^{c_{ij}^{(1)+}u_{pi}^{+}+0u_{pi}^{-}}\Big{|}x_{ij}^{c_{ij}^{(1)+}0+0u_{i}^{-}}=1$.
Thus we have $u_{pi}^{+}=0$. For $ik$ such that $c_{ik}^{(1)}<0$ the relation
$x_{ik}^{c_{ik}^{(1)+}u_{pi}^{+}+c_{ik}^{(1)-}u_{pi}^{-}}\Big{|}x_{ik}^{c_{ik}^{(1)+}u_{i}^{+}+c_{ik}^{(1)-}u_{i}^{-}}$
implies
$x_{ik}^{0u_{pi}^{+}+c_{ik}^{(1)-}u_{pi}^{-}}\Big{|}x_{ik}^{0u_{i}^{+}+c_{ik}^{(1)-}u_{i}^{-}}$,
and then $u_{pi}^{-}\leq u_{i}^{-}$. Summing up, we have obtained that
$u_{i}=-u_{i}^{-}\leq-u_{pi}^{-}=u_{pi}\leq 0$. Therefore
$y_{it}^{c_{it}^{(2)+}u_{pi}^{+}+c_{it}^{(2)-}u_{pi}^{-}}\Big{|}y_{it}^{c_{it}^{(2)+}u_{i}^{+}+c_{it}^{(2)-}u_{i}^{-}}$
and thus $y_{it}^{(D_{2}({\bf u_{p}})^{+})_{it}}\big{|}y_{it}^{(D_{2}({\bf
u})^{+})_{it}}$.

From all of the above cases we conclude that $y^{D_{2}({\bf
u_{p}})^{+}}\big{|}y^{D_{2}({\bf u})^{+}}$. Therefore $y^{D_{2}({\bf
u})^{+}}\in(y^{D_{2}({\bf u_{1}})^{+}},\ldots,y^{D_{2}({\bf u_{r}})^{+}})$ and
thus $\mathcal{G}^{\prime}$ is a Gröbner basis of $I_{A_{2}}$ with respect to
$<_{2}$. Finally, to prove that $\mathcal{G}^{\prime}$ is reduced we argue by
contradiction. This implies that there exists an integer $p$ such that
$y^{D_{2}({\bf u_{p}})^{-}}\in\operatorname{in}_{<_{2}}(I_{A_{2}})$, so
$y^{D_{2}({\bf u_{p}})^{-}}$ is divisible by some $y^{D_{2}({\bf
u_{j}})^{+}}$. This in turn, by a similar argument like the division argument
before, implies that $x^{D_{1}({\bf u_{j}})^{+}}|x^{D_{1}({\bf u_{p}})^{-}}$,
a contradiction since $\mathcal{G}$ is reduced. Therefore we obtain that
$\mathcal{G}^{\prime}$ is a reduced Gröbner basis with respect to $<_{2}$, as
desired. It follows that $f:D_{1}(\textbf{u})\mapsto D_{2}(\bf{u})$ is a one-
to-one correspondence between the universal Gröbner bases of $I_{A_{1}}$ and
$I_{A_{2}}$.

Note that every reduced Gröbner basis is also a Markov basis, not necessarily
minimal, thus the set of indispensables elements of $I_{A}$ is a subset of the
intersection of all the reduced Gröbner bases. Note that the set of
indispensables from [16, Theorem 2.4] is the intersection of the lexicographic
reduced Gröbner bases or from [17, Theorem 13] is the intersection of the
degree reverse lexicographic reduced Gröbner bases. Therefore the set of
indispensables elements of $I_{A}$ is the intersection of all reduced Gröbner
bases. This observation and Theorem 2.3 provides a second proof of the fact
that there exists a one-to-one correspondence between the indispensable
binomials of the two $T_{\omega}$-robust toric ideals $I_{A_{1}}$ and
$I_{A_{2}}$.

##  3\. Strongly Robust Ideals

A binomial ${\bf x}^{{\bf u}^{+}}-{\bf x}^{{\bf u}^{-}}\in I_{A}$ is called
primitive if there is no other binomial ${\bf x}^{{\bf v}^{+}}-{\bf x}^{{\bf
v}^{-}}\in I_{A}$ such that ${\bf x}^{{\bf v}^{+}}$ divides ${\bf x}^{{\bf
u}^{+}}$ and ${\bf x}^{{\bf v}^{-}}$ divides ${\bf x}^{{\bf u}^{-}}$. The set
of the primitive binomials is finite, it is called the Graver basis of $I_{A}$
and is denoted by $\operatorname{Gr}(I_{A})$, [22, Chapter 4]. By
$\operatorname{Gr}(A)$ we denote the set $\\{{\bf u}|{\bf x}^{{\bf
u}^{+}}-{\bf x}^{{\bf u}^{-}}\in\operatorname{Gr}(I_{A})\\}$, it is the set of
elements in $\operatorname{Ker}_{\mathbb{Z}}(A)$ that do not have a proper
conformal decomposition. An element ${\bf
u}\in\operatorname{Ker}_{\mathbb{Z}}(A)$ has a conformal decomposition ${\bf
u}={\bf v}+_{c}{\bf w}$ if it can be written in the form ${\bf v}+{\bf w}$,
where ${\bf v},{\bf w}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$ and ${\bf
u}^{+}={\bf v}^{+}+{\bf w}^{+}$, ${\bf u}^{-}={\bf v}^{-}+{\bf w}^{-}$. The
conformal decomposition is called proper if both ${\bf v}$ and ${\bf w}$ are
not zero.

A strongly robust toric ideal is a toric ideal $I_{A}$ for which the Graver
basis $\operatorname{Gr}(I_{A})$ is a minimal system of generators.
Furthermore, since the ideal $I_{A}$ is positively graded, then any minimal
system of binomial generators is a subset of the Graver basis, see for example
[8, Theorem 2.3], and therefore there is only one minimal system of
generators, which is the Graver basis. We conclude that, in a strongly robust
toric ideal, every Graver basis element is indispensable. As was explained at
the end of Section 2, the set of indispensable elements is the intersection of
all reduced Gröbner bases, which is a subset of the Universal Gröbner basis,
which at its own is a subset of the Graver basis (see [22, Chapter 4]). Thus,
we conclude that for a strongly robust toric ideal $I_{A}$, the following sets
are identical: the set of indispensable elements, any minimal system of
generators, any reduced Gröbner basis, the Universal Gröbner basis and the
Graver basis.

The following results will enable us to define a simplicial complex, the
strongly robustness complex, which determines the strongly robustness property
for toric ideals. According to Theorem 2.2, for any two $T_{\omega}$-robust
toric ideals $I_{A_{1}},I_{A_{2}}$, we have that $D_{1}({\bf u})$ is an
indispensable element of $A_{1}$ if and only if $D_{2}({\bf u})$ is an
indispensable element of $A_{2}$, for an element ${\bf
u}\in\operatorname{Ker}_{\mathbb{Z}}(T)$. Actually ${\bf
u}\in\operatorname{Gr}(T)$, since indispensable elements of $I_{A_{1}}$ are in
the Graver basis of $I_{A_{1}}$ and there is a one-to-one correspondence
between $\operatorname{Gr}(I_{T})$ and $\operatorname{Gr}(I_{A_{1}})$, given
by the map ${\bf u}\mapsto D_{1}({\bf u})$, see [19, Theorem 1.11]. Therefore,
the set of elements ${\bf u}$ which belong to $\operatorname{Gr}(T)$, such
that $D({\bf u})$ is indispensable in a $T_{\omega}$-robust toric ideal
$I_{A}$, does not depend on the $I_{A}$ chosen, but only on $T$ and $\omega$.
Thus, the following definition is meaningful.

######  Definition 3.1.

Let $I_{T}$ be a simple toric ideal and $\omega\subseteq[s]$. We denote by

$S_{\omega}(T)=\\{{\bf u}\in\operatorname{Gr}(T)|\ D({\bf u})\in S(A)\\}$

and call $S_{\omega}(T)$ the $T_{\omega}$-indispensable set, where $I_{A}$ is
an $T_{\omega}$-robust toric ideal and $S(A)$ is the set of indispensable
elements of $A$.

In the next example we compute several sets $S_{\omega}(T)$ for different
$\omega$’s. It is easy to observe an important property of the sets
$S_{\omega}(T)$, displayed in Proposition 3.3, that will help us introduce a
simplicial complex on the set $[s]$.

######  Example 3.2.

Let $I_{T}$ be the simple toric ideal with $T=(7\ 8\ 9)$. Then, using $4ti2$
[1], we obtain that $\operatorname{Gr}(T)$ has the following 11 elements:
$(1,-2,1)$, $(5,-1,-3)$, $(4,1,-4)$, $(0,9,-8)$, $(1,7,-7)$, $(2,5,-6)$,
$(3,3,-5)$, $(9,0,-7)$, $(6,-3,-2)$, $(8,-7,0)$, $(7,-5,-1)$. For each subset
$\omega\subseteq[3]$ we compute with $4ti2$ [1] a Markov basis for one example
of a $T_{\omega}$-robust ideal. Using the simple Algorithm 1 of [9, Section 3]
we compute the indispensable elements for each example. Then, Theorem 2.2
gives us that the $T_{\omega}$-indispensable sets for different $\omega$’s
are:
$S_{\emptyset}(T)=\operatorname{Gr}(T)$,
$S_{\\{1\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4),(9,0,-7),(6,-3,-2),(7,-5,-1),\\\
(8,-7,0)\\}$,
$S_{\\{2\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4),(0,9,-8),(1,7,-7),(2,5,-6),(3,3,-5),\\\
(6,-3,-2),(7,-5,-1),(8,-7,0)\\}$,
$S_{\\{3\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4),(0,9,-8),(1,7,-7),(2,5,-6),(3,3,-5),\\\
(9,0,-7)\\}$,
$S_{\\{1,2\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4),(6,-3,-2),(7,-5,-1),(8,-7,0)\\}$,
$S_{\\{1,3\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4),(9,0,-7)\\}$,
$S_{\\{2,3\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4),(0,9,-8),(1,7,-7),(2,5,-6),\\\
(3,3,-5)\\}$ and $S_{\\{1,2,3\\}}(T)=\\{(1,-2,1),(5,-1,-3),(4,1,-4)\\}.$

For the next proof we will exploit the notion of semiconformal decomposition
which characterizes the indispensable elements of toric ideals (see [15, Lemma
3.10] and [7, Proposition 1.1]). We recall from [15, Definition 3.9] that for
vectors ${\bf u},{\bf v},{\bf w}\in\operatorname{Ker}_{\mathbb{Z}}(A)$ such
that ${\bf u}={\bf v}+{\bf w}$, the sum is said to be a semiconformal
decomposition of ${\bf u}$, written ${\bf u}={\bf v}+_{sc}{\bf w}$, if
$v_{i}>0$ implies that $w_{i}\geq 0$, and $w_{i}<0$ implies that $v_{i}\leq
0$, for all $1\leq i\leq n$. The decomposition is called proper if both ${\bf
v},{\bf w}$ are nonzero. The set of indispensable elements
${{\mathcal{S}}}(A)$ of $A$ consists of all nonzero vectors in
$\operatorname{Ker}_{{\mathbb{Z}}}({A})$ with no proper semiconformal
decomposition, see [7, Proposition 1.1].

######  Proposition 3.3.

Let $I_{T}\subset K[x_{1},\dots,x_{s}]$ be a simple toric ideal and
$\omega_{1}\subseteq\omega_{2}\subseteq[s]$. Then we have that
$S_{\omega_{1}}(T)\supseteq S_{\omega_{2}}(T)$.

###### Proof.

Consider the $T_{\omega_{2}}$-robust toric ideal $I_{A_{2}}$. According to
Definition 3.1, we have that the bouquet ideal of $I_{A_{2}}$ is $I_{T}$ and
$\omega_{2}=\\{i\in[n]|\ {B_{i}^{(2)}}\ \text{is non-mixed}\\}.$ Consider now
a subset $\omega_{1}\subseteq\omega_{2}$. Similarly, from Definition 3.1, the
bouquet ideal of a $T_{\omega_{1}}$-robust toric ideal $I_{A_{1}}$ is $I_{T}$
and $\omega_{1}=\\{i\in[n]|\ {B_{i}^{(1)}}\ \text{is non-mixed}\\}.$

Let ${\bf u}\in S_{\omega_{2}}(T)$, our goal is to show that ${\bf u}\in
S_{\omega_{1}}(T)$. According to the Definition 3.1, applied for
$S_{\omega_{2}}(T)$, we have $D_{2}({\bf u})\in S(A_{2})$. Since $D_{2}({\bf
u})\in\operatorname{Ker}_{\mathbb{Z}}(A_{2})$, from the one-to-one
correspondence between the elements of $\operatorname{Ker}_{\mathbb{Z}}(T)$
and the elements of each of $\operatorname{Ker}_{\mathbb{Z}}(A_{1})$ and
$\operatorname{Ker}_{\mathbb{Z}}(A_{2})$ (see [19, Theorem 1.9]), we have that
$D_{1}({\bf u})\in\operatorname{Ker}_{\mathbb{Z}}(A_{1})$ as well.

Suppose that $D_{1}({\bf u})=D_{1}({\bf v})+_{sc}D_{1}({\bf w})$ is a
semiconformal decomposition of $D_{1}({\bf u})$, where $D_{1}({\bf
v}),D_{1}({\bf w})\in\operatorname{Ker}_{\mathbb{Z}}(A_{1})$. For each
$i=1,\ldots,s$, at coordinate level we have that

$\left(c^{(1)}_{i1},c^{(1)}_{i2},\ldots
c^{(1)}_{ik_{i}}\right)u_{i}=\left(c^{(1)}_{i1},c^{(1)}_{i2},\ldots
c^{(1)}_{ik_{i}}\right)v_{i}+_{sc}\left(c^{(1)}_{i1},c^{(1)}_{i2},\ldots
c^{(1)}_{ik_{i}}\right)w_{i},$

where we know that all $c^{(1)}_{i1}$ are positive, while the $c^{(1)}_{ij}$
are all positive if the bouquet $B^{(1)}_{i}$ is non-mixed, and some of the
$c^{(1)}_{ij}$ are negative if the bouquet $B^{(1)}_{i}$ is mixed.

For all $i$, we have from
$c^{(1)}_{i1}u_{i}=c^{(1)}_{i1}v_{i}+_{sc}c^{(1)}_{i1}w_{i}$ that the sum
$u_{i}=v_{i}+_{sc}w_{i}$ of the $i$-th coordinates is a semiconformal sum,
since all $c^{(1)}_{i1}>0$. For $i\not\in\omega_{1}$, we have a stronger
relation, namely that the sum $u_{i}=v_{i}+_{sc}w_{i}$ of the $i$-th
coordinates is a conformal sum. Indeed, if $B^{(1)}_{i}$ is free we have
$u_{i}=v_{i}=w_{i}=0$ and the sum $0=0+0$ is conformal. Otherwise, if
$B^{(1)}_{i}$ is mixed we distinguish three cases.

Case 1: If $u_{i}>0$, then $c^{(1)}_{i1}u_{i}>0$ and the sum being
semiconformal implies that $c^{(1)}_{i1}w_{i}\geq 0\Rightarrow w_{i}\geq 0$.
Since $B^{(1)}_{i}$ is mixed at least one of the $c^{(1)}_{ij}$, for $j\geq 2$
is negative, in which case $c^{(1)}_{ij}u_{i}<0$ and from the semiconformality
we have that $c^{(1)}_{ij}v_{i}\leq 0\Rightarrow v_{i}\geq 0$. Therefore we
proved that

$u_{i}>0\Rightarrow v_{i}\geq 0\text{ and }w_{i}\geq 0.$

Case 2: If $u_{i}<0$, then $c_{i1}^{(1)}u_{i}<0$ and the sum being
semiconformal implies that $c^{(1)}_{i1}v_{i}\leq 0\Rightarrow v_{i}\leq 0$.
As before, since $B^{(1)}_{i}$ is mixed, at least one $c_{ij}^{(1)}$ is
negative, and thus $c^{(1)}_{ij}u_{i}>0$. It follows now from the
semiconformality that $c^{(1)}_{ij}w_{i}\geq 0\Rightarrow w_{i}\leq 0$,
therefore

$u_{i}<0\Rightarrow v_{i}\leq 0\text{ and }w_{i}\leq 0.$

Case 3: Finally, if $u_{i}=0$, then $c^{(1)}_{i1}u_{i}=0$ and the sum being
semiconformal implies that $c^{(1)}_{i1}v_{i}\leq 0\text{ and
}c^{(1)}_{i1}w_{i}\geq 0$. Since $c^{(1)}_{i1}>0$, we have that $v_{i}\leq
0\text{ and }w_{i}\geq 0$. Since $B^{(1)}_{i}$ is mixed, there exists a
negative $c^{(1)}_{ij}$, and we know from the semiconformality that
$c^{(1)}_{ij}v_{i}\leq 0\text{ and }c^{(1)}_{ij}w_{i}\geq 0$, which implies
that $v_{i}\geq 0\text{ and }w_{i}\leq 0$. Again we have proved that

$u_{i}=0\Rightarrow v_{i}=0\text{ and }w_{i}=0\text{.}$

The three cases analyzed before imply that for $i\not\in\omega_{1}$ the sum
$u_{i}=v_{i}+w_{i}$ of the $i$-th coordinates is a conformal sum.

For $i\in\omega_{2}$, the sum $u_{i}=v_{i}+_{sc}w_{i}$ is semiconformal and
stays semiconformal if it is multiplied with a positive $c^{(2)}_{ij}$. For
$i\not\in\omega_{2}$, we have that $i\not\in\omega_{1}$, since
$\omega_{1}\subseteq\omega_{2}$, therefore the sum $u_{i}=v_{i}+w_{i}$ is
conformal and remains conformal if it is multiplied by a positive or a
negative $c^{(2)}_{ij}$. Since conformality implies semiconformality then
$D_{2}({\bf u})=D_{2}({\bf v})+_{sc}D_{2}({\bf w})$, and $D_{2}({\bf u})\in
S(A_{2})$. Thus either $D_{2}({\bf v})={\bf 0}$ or $D_{2}({\bf w})={\bf 0}$,
since indispensable elements do not have proper semiconformal decomposition
[7, Proposition 1.1]. But this in turn gives that either ${\bf v}={\bf 0}$ or
${\bf w}={\bf 0}$ which implies that either $D_{1}({\bf v})={\bf 0}\text{ or
}D_{1}({\bf w})={\bf 0}$. This means that $D_{1}({\bf u})\in S(A_{1})$ or
equivalently ${\bf u}\in S_{\omega_{1}}(T)$, as desired.

######  Definition 3.4.

The set $\Delta_{T}=\\{\omega\subseteq[s]\ |\
S_{\omega}(T)=\operatorname{Gr}(T)\\}$ is called the strongly robustness
complex of $T$.

######  Corollary 3.5.

The strongly robustness complex of $T$ is a simplicial complex.

###### Proof.

Suppose that $\omega\in\Delta_{T}$ and let $\omega^{\prime}\subseteq\omega$.
Then $\operatorname{Gr}(T)=S_{\omega}(T)\subseteq
S_{\omega^{\prime}}(T)\subseteq\operatorname{Gr}(T)$, by Proposition 3.3. Thus
$S_{\omega}=\operatorname{Gr}(T)$ and $\omega^{\prime}\in\Delta_{T}$.

In the Example 3.2 the strongly robustness complex $\Delta_{T}$ is the empty
set. In the case that all bouquets are mixed, Corollary 4.4 of [19] states
that the ideal is strongly robust, thus
$S_{\emptyset}(T)=\operatorname{Gr}(T)$ and therefore
$\emptyset\in\Delta_{T}$. This means that $\Delta_{T}$ is never the void
simplicial complex.

Boocher and Robeva posed the problem of describing combinatorially the
robustness property for toric ideals, see [4, Question 3.5]. According to
Theorem 3.6 below, the simplicial complex $\Delta_{T}$ determines the strongly
robustness property for toric ideals. However, it is still an open problem
whether the robustness property of toric ideals is equivalent to the strongly
robustness property, see Section 4 for more details and results.

######  Theorem 3.6.

Let $I_{A}$ be a $T_{\omega}$-robust toric ideal. The toric ideal $I_{A}$ is
strongly robust if and only if $\omega$ is a face of the strongly robustness
complex $\Delta_{T}$.

###### Proof.

A toric ideal $I_{A}$ is strongly robust if and only if every Graver basis
element is indispensable. Thus for the $T_{\omega}$-robust ideal $I_{A}$ we
have: $I_{A}$ is strongly robust if and only if
$S(I_{A})=\operatorname{Gr}(I_{A})$ if and only if
$S_{\omega}(T)=\operatorname{Gr}(T)$ if and only if $\omega\in\Delta_{T}$.

The following theorem provides a way to compute the strongly robustness
complex of a simple toric ideal $I_{T}$. By $\Lambda(T)$ we denote the second
Lawrence lifting of $T$, which is the $(m+s)\times 2s$ matrix
${\begin{pmatrix}T&0\\\ I_{s}&I_{s}\end{pmatrix}}.$ By $\Lambda(T)_{\omega}$
we denote the matrix taken from $\Lambda(T)$ by removing the $(m+i)$-th row
and the $(s+i)$-th column for each $i\in\omega$. For example, for
$T={\footnotesize\begin{pmatrix}12&9&8&0\\\ 0&3&4&12\end{pmatrix}}$ and
$\omega=\\{2,4\\}$, we have that

$\Lambda(T)={\footnotesize\begin{pmatrix}12&9&8&0&0&0&0&0\\\
0&3&4&12&0&0&0&0\\\ 1&0&0&0&1&0&0&0\\\ 0&1&0&0&0&1&0&0\\\ 0&0&1&0&0&0&1&0\\\
0&0&0&1&0&0&0&1\\\ \end{pmatrix}}\ \ \text{ and }\ \
\Lambda(T)_{\omega}={\footnotesize\begin{pmatrix}12&9&8&0&0&0\\\
0&3&4&12&0&0\\\ 1&0&0&0&1&0\\\ 0&0&1&0&0&1\\\ \end{pmatrix}}.$

######  Theorem 3.7.

The set $\omega$ is a face of the strongly robustness complex $\Delta_{T}$ if
and only if $\Lambda(T)_{\omega}$ is strongly robust.

###### Proof.

Note that if ${\bf b}_{1},\ldots,{\bf b}_{s}$ are the row vectors of the Gale
transform of $T$ then ${\bf b}_{1},\ldots,{\bf
b}_{s},-\textbf{b}_{1},\ldots,-{\bf b}_{s}$ are the row vectors of the Gale
transform of $\Lambda(T)$ and ${\bf b}_{1},\ldots,{\bf b}_{s}$, and $-{\bf
b}_{j}$ for $j\not\in\omega$ are the row vectors of the Gale transform of
$\Lambda(T)_{\omega}$. Therefore $\Lambda(T)_{\omega}$ has $s$ bouquets of the
form: $\\{{\bf a}_{i}\\}$ for $i\in\omega$, and $\\{{\bf a}_{j},{\bf
a}_{j+s}\\}$ for $j\not\in\omega$, where the indices are the indices of the
columns of $\Lambda(T)$. The former are non-mixed and the later are free if
${\bf b}_{j}={\bf 0}$ or mixed, since ${\bf b}_{j+s}=-{\bf b}_{j}$. This means
that $I_{\Lambda(T)_{\omega}}$ is a $T_{\omega}$-robust toric ideal. Then
$\omega\in\Delta_{T}$ if and only if $\Lambda(T)_{\omega}$ is strongly robust,
by Theorem 3.6.

It follows also that the strongly robustness simplicial complex depends only
on the simple toric ideal $I_{T}$.

######  Example 3.8.

Theorem 3.7 provides a way to construct strongly robust ideals. Consider the
$2\times 4$ integer matrix

$T={\footnotesize\begin{pmatrix}12&9&8&0\\\ 0&3&4&12\end{pmatrix}}.$

Computing the Gale transform one can easily see that $I_{T}$ is a simple toric
ideal, see Example 4.3.

To find the vertices of the simplicial complex one first computes, e.g. using
4ti2 [1], minimal Markov and Graver bases of $\Lambda(T)_{\\{i\\}}$ for all
$i\in[4]$. It turns out that $\Lambda(T)_{\\{2\\}}$, $\Lambda(T)_{\\{4\\}}$
are strongly robust, since in both cases the Graver basis and a minimal Markov
basis given by 4ti2 are identical, while $\Lambda(T)_{\\{1\\}}$ and
$\Lambda(T)_{\\{3\\}}$ are not strongly robust. Thus $2$ and $4$ are the only
vertices of $\Delta_{T}$. Finally, in this example, one has to check whether
$\\{2,4\\}$ belongs or not to $\Delta_{T}$. It turns out that
$\Lambda(T)_{\\{2,4\\}}$ is strongly robust, thus, the simplicial complex
$\Delta_{T}$ is $\\{\emptyset,\\{2\\},\\{4\\},\\{2,4\\}\\}$. From Theorem 3.6,
all $T_{\emptyset},T_{\\{2\\}},T_{\\{4\\}},T_{\\{2,4\\}}$-robust toric ideals
are strongly robust, while for instance none of the $T_{\\{3\\}}$ or
$T_{\\{1,3,4\\}}$-robust toric ideals are strongly robust. To construct a
strongly robust ideal take any $\omega\in\Delta_{T}$, say $\\{2,4\\}$, and
choose four primitive integer vectors in such a way that the first and the
third are mixed while the second and the fourth are not mixed. For example,
say ${\bf c}_{1}=(1,-1),{\bf c}_{2}=(5,6,7,8),{\bf c}_{3}=(2023,-2022,11),{\bf
c}_{4}=(13,14,15)$. Computing any integer solution of the equation
$1=\lambda_{i1}c_{i1}+\cdots+\lambda_{im_{i}}c_{im_{i}}$ for each vector ${\bf
c}_{i}=(c_{i1},c_{i2},\ldots,c_{im_{i}})$, for example $1=1\cdot
1+0\cdot(-1)$, $1=-1\cdot 5+1\cdot 6+0\cdot 7+0\cdot 8$, $1=1\cdot
2023+1\cdot(-2022)+0\cdot(11)$, $1=0\cdot 13+(-1)\cdot 14+1\cdot 15$, we get
the following generalized Lawrence matrix

$\tiny{D=\ \left(\begin{array}[]{cccccccccccc}12&0&-9&9&0&0&8&8&0&0&0&0\\\
0&0&-3&3&0&0&4&4&0&0&-12&12\\\ 1&1&0&0&0&0&0&0&0&0&0&0\\\
0&0&-6&5&0&0&0&0&0&0&0&0\\\ 0&0&-7&0&5&0&0&0&0&0&0&0\\\
0&0&-8&0&0&5&0&0&0&0&0&0\\\ 0&0&0&0&0&0&2022&2023&0&0&0&0\\\
0&0&0&0&0&0&-11&0&2023&0&0&0\\\ 0&0&0&0&0&0&0&0&0&-14&13&0\\\
0&0&0&0&0&0&0&0&0&-15&0&13\end{array}\right)\in{\mathbb{Z}}^{10\times 12},}$

for more details see [19, Section 2]. Then $I_{D}$ is a $T_{\\{2,4\\}}$-robust
ideal, therefore it is strongly robust by Theorem 3.6, since
$\\{2,4\\}\in\Delta_{T}$.

In fact, Theorem 3.6 implies that all strongly robust ideals are taken in this
way since: 1) every toric ideal is a $T_{\omega}$-robust ideal for some $T$
and $\omega$, and 2) all toric ideals have a representation in the form
$I_{A^{\prime}}$ where $A^{\prime}$ is a generalized Lawrence matrix, see [19,
Corollary 2.3].

##  4\. The codimension two case

Let $A$ be an integer matrix $m\times n$ of rank $n-2$ such that the toric
ideal $I_{A}$ is positively graded. Then the columns of $A$ form an acyclic
vector configuration, see [29, Section 6.2]. Let $\\{{\bf b}_{1},{\bf
b}_{2},\dots,{\bf b}_{n}\\}$ be the Gale transform of $A$, where ${\bf
b}_{i}=(b_{i1},b_{i2})$ for $i=1,\ldots,n$, and $B$ the $n\times 2$ matrix
with rows ${\bf b}_{1},{\bf b}_{2},\dots,{\bf b}_{n}$. Let
$\widetilde{B}=\\{{\widetilde{\bf b}_{1}},{\widetilde{\bf
b}_{2}},\dots,{\widetilde{\bf b}_{n}}\\}$ be the reduced Gale transform of
$A$, the set where ${\widetilde{\bf
b}_{i}}=gcd(b_{i1},b_{i2})^{-1}(-b_{i2},b_{i1})$ if ${\bf b}_{i}\not=(0,0)$,
and ${\widetilde{\bf b}_{i}}=(0,0)$ if ${\bf b}_{i}=(0,0)$. Note that ${\bf
b}_{i}=(0,0)$ means that ${\bf a}_{i}$ is a free vector. Since both
configurations are totally cyclic, see [29, Section 6.4], we can reorder all
the nonzero vectors ${\bf b}_{1},{\bf b}_{2},\ldots,{\bf b}_{n^{\prime}}$,
$n^{\prime}\leq n$, in such a way that in the interior of the
$\operatorname{cone}({\widetilde{\bf b}_{i}},{\widetilde{\bf b}_{i+1}})$ there
is no other ${\widetilde{\bf b}_{j}}$ for all $i=1,\ldots,n$, where
${\widetilde{\bf b}_{n^{\prime}+1}}={\widetilde{\bf b}_{1}}$. The minimal
generating set of the monoid $\operatorname{cone}({\widetilde{\bf
b}_{i}},{\widetilde{\bf b}_{i+1}})\cap\mathbb{Z}^{2}$ is the Hilbert basis of
the $\operatorname{cone}({\widetilde{\bf b}_{i}},{\widetilde{\bf b}_{i+1}})$
and is denoted by $H_{i}$.

######  Theorem 4.1.

[24, Corollary 2.5] [18, Theorem 3.7] Let $A\in\mathbb{Z}^{(n-2)\times n}$ of
rank $n-2$ and $Ker_{\mathbb{Z}}(A)\cap\mathbb{N}^{n}=\\{{\bf 0}\\}.$

  * •

A vector ${\bf u}\in\mathbb{Z}^{2}$ has either ${\bf u}$ or $-{\bf u}$ in
$H_{1}\cup H_{2}\cup\cdots\cup H_{n^{\prime}}$ if and only if $x^{{\bf
u}^{+}}-x^{{\bf u}^{-}}$ is in the Graver basis of $I_{A}$.

  * •

A vector ${\bf u}\in\mathbb{Z}^{2}$ has both ${\bf u}$ and $-{\bf u}$ in
$H_{1}\cup H_{2}\cup\cdots\cup H_{n^{\prime}}$ if and only if $x^{{\bf
u}^{+}}-x^{{\bf u}^{-}}$ is an indispensable element of $I_{A}$.

Let $I_{A}$ be a $T_{\omega}$-robust ideal of codimension 2, and
${\widetilde{B}}=\\{{\widetilde{\bf b}_{1}},\ldots,{\widetilde{\bf b}_{n}}\\}$
be the reduced Gale transform of $A$ and $\\{{\widetilde{\bf
t}_{1}},\ldots,{\widetilde{\bf t}_{s}}\\}$ be the reduced Gale transform of
$T$. Using the isomorphism $D$ from $\operatorname{Ker}_{{\mathbb{Z}}}({T})$
to $\operatorname{Ker}_{{\mathbb{Z}}}(A)$, we get that if ${\bf
t}_{i}=(t_{i1},t_{i2})$ then for an ${\bf a}_{j}$ that is the $k$-th element
of the bouquet $B_{i}$ we have ${\bf b}_{j}=(c_{ik}t_{i1},c_{ik}t_{i2})$. If
$c_{ik}$ is positive then ${\widetilde{\bf b}_{j}}={\widetilde{\bf t}_{i}}$,
and if $c_{it}$ is negative then ${\widetilde{\bf b}_{j}}=-{\widetilde{\bf
t}_{i}}$. Thus, if ${\bf a}_{j}$ belongs to a non-mixed or free bouquet
$B_{i}$, then ${\widetilde{\bf b}_{j}}={\widetilde{\bf t}_{i}}$. For a mixed
bouquet $B_{i}$, we have ${\widetilde{\bf b}_{j}}={\widetilde{\bf t}_{i}}$ for
at least one ${\bf a}_{j}\in B_{i}$ and we also have ${\widetilde{\bf
b}_{k}}=-{\widetilde{\bf t}_{i}}$ for at least one ${\bf a}_{k}\in B_{i}$.

######  Theorem 4.2.

Let $I_{T}$ be a simple codimension 2 toric ideal. Then the strongly
robustness complex $\Delta_{T}$ is a simplicial subcomplex of the simplex with
vertices $i\in[s]$, such that $\widetilde{\bf t}_{i}$ is not a vertex of the
centrally symmetric polygon

$P=\operatorname{conv}\left(\left\\{\widetilde{\bf t}_{i},-\widetilde{\bf
t}_{i}\big{|}\ i\in[s]\right\\}\right).$

###### Proof.

Let $I_{A}$ be a $T_{\omega}$-robust ideal,
${\widetilde{B}}=\\{{\widetilde{\bf b}_{1}},\ldots,{\widetilde{\bf b}_{n}}\\}$
be the reduced Gale transform of $A$ and ${\widetilde{T}}=\\{{\widetilde{\bf
t}_{1}},\ldots,{\widetilde{\bf t}_{s}}\\}$ be the reduced Gale transform of
$T$. Then, as explained before, the set ${\widetilde{B}}$ is equal to
$\\{{\widetilde{\bf t}_{1}},\ldots,{\widetilde{\bf
t}_{s}}\\}\cup\\{-{\widetilde{\bf t}_{i}}|\ i\not\in\omega\\}$. Then we claim
that $\operatorname{conv}({\widetilde{B}})$ is centrally symmetric if and only
if $P=\operatorname{conv}({\widetilde{B}})$. Indeed, if
$P=\operatorname{conv}({\widetilde{B}})$, then
$\operatorname{conv}({\widetilde{B}})$ is centrally symmetric since $P$ is
centrally symmetric. Conversely, if $\operatorname{conv}({\widetilde{B}})$ is
centrally symmetric, then for each ${\widetilde{\bf b}_{j}}\in{\widetilde{B}}$
we have $-{\widetilde{\bf b}_{j}}\in\operatorname{conv}({\widetilde{B}})$.
Thus, all ${\widetilde{\bf t}_{i}}$ and $-{\widetilde{\bf t}_{i}}$ are in
$\operatorname{conv}({\widetilde{B}})$, and
$P=\operatorname{conv}(\\{\widetilde{\bf t}_{i},-\widetilde{\bf t}_{i}|\
i\in[s]\\})$ therefore

$P\subseteq\operatorname{conv}({\widetilde{B}})=\operatorname{conv}\left(\left\\{{\widetilde{\bf
t}_{1}},\ldots,{\widetilde{\bf t}_{s}}\right\\}\cup\left\\{-{\widetilde{\bf
t}_{i}}\big{|}i\not\in\omega\right\\}\right)\subseteq\operatorname{conv}\left(\left\\{\widetilde{\bf
t}_{i},-\widetilde{\bf t}_{i}\big{|}i\in[s]\right\\}\right)=P,$

thus $P=\operatorname{conv}({\widetilde{B}})$.

Let $\omega\in\Delta_{T}$. Then $S_{\omega}(T)=\operatorname{Gr}(T)$, and
therefore $I_{A}$ is strongly robust. Using [24, Theorem 1.2] we obtain that
$\operatorname{conv}({\widetilde{B}})$ is a centrally symmetric polygon, and
thus $P=\operatorname{conv}({\widetilde{B}})$. We conclude that
$P=\operatorname{conv}\left({\widetilde{B}}=\left\\{{\widetilde{\bf
t}_{1}},\ldots,{\widetilde{\bf t}_{s}}\right\\}\cup\left\\{-{\widetilde{\bf
t}_{i}}\big{|}i\not\in\omega\right\\}\right)$, which implies that for
$i\in\omega$, $\widetilde{\bf t}_{i}$ is not a vertex of the centrally
symmetric polygon $P$.

######  Example 4.3.

Theorem 4.2 provides an alternative way to the one developed in Section 3 to
construct the strongly robustness simplicial complex for codimension 2 toric
ideals. Consider again the $2\times 4$ integer matrix

$T={\footnotesize\begin{pmatrix}12&9&8&0\\\ 0&3&4&12\end{pmatrix}},$

which has the following Gale transform $B$, and whose matrix associated to the
reduced Gale transform, denoted by $\widetilde{B}$, respectively,

$B={\footnotesize\begin{pmatrix}2&3\\\ 0&-4\\\ -3&0\\\ 1&1\end{pmatrix}}\ \ \
\text{ and }\ \ \ \widetilde{B}={\footnotesize\begin{pmatrix}-3&2\\\ 1&0\\\
0&-1\\\ -1&1\end{pmatrix}}.$

Then $I_{T}$ is a simple toric ideal, since no two column vectors are
parallel, and the convex hull of the set $\\{\widetilde{\bf
b}_{i},-\widetilde{\bf b}_{i}|\ i\in[s]\\}$ is a parallelogram $P$ with
vertices $\widetilde{\bf b}_{1}=(-3,2)$, $-\widetilde{\bf b}_{1}=(3,-2)$,
$-\widetilde{\bf b}_{3}=(0,1)$, $\widetilde{\bf b}_{3}=(0,-1)$. According to
Theorem 4.2, the geometry of this polygon influences the simplicial complex
$\Delta_{T}$, which is a subcomplex of
$\\{\emptyset,\\{2\\},\\{4\\},\\{2,4\\}\\}$, since $\widetilde{\bf
b}_{2},\widetilde{\bf b}_{4}$ are not vertices of $P$. However, by [24,
Proposition 2.4], $-\widetilde{\bf b}_{2}=(-1,0)$ is in the Hilbert basis of
the $\operatorname{cone}(\widetilde{\bf b}_{1},\widetilde{\bf b}_{3})$ and
$-\widetilde{\bf b}_{4}=(1,-1)$ is in the Hilbert basis of the
$\operatorname{cone}(\widetilde{\bf b}_{3},-\widetilde{\bf b}_{1})$, both of
which do not involve the indices $2,4$. Therefore, all the sets
$\\{2\\},\\{4\\},\\{2,4\\}$ are faces of $\Delta_{T}$. Thus,
$\Delta_{T}=\\{\emptyset,\\{2\\},\\{4\\},\\{2,4\\}\\}$.

Finally, an ideal is called robust if the universal Gröbner basis is a minimal
generating set. It follows from [26, Theorem 5.10] that the universal Gröbner
basis is the set of indispensables, and thus every strongly robust toric ideal
is robust. In [5], Boocher et al. proved that every robust toric ideal
associated to a graph is strongly robust and they asked if for every toric
ideal robust implies strongly robust. In Theorem 4.5, we prove that this is
true for codimension 2 toric ideals. Note that for toric ideals of codimension
1 this is obvious, since they are principal. A key result for the proof is
that circuits are always included in the Universal Gröbner basis, see [22,
Proposition 4.11]. We recall that circuits are irreducible binomials with
minimal support.

######  Lemma 4.4.

Let $I_{A}$ be a codimension 2 toric ideal. Then $C$ is a circuit if and only
if there exists an $i$ such that $C=x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf b}^{t}_{i}})}^{-}}$, where $B$ is the
Gale transform of $A$.

###### Proof.

Let $C$ be a circuit of $I_{A}$. Then $C\in I_{A}$ and has no monomial factor,
therefore it is in the form $x^{(B{{\bf u}^{t}})^{+}}-x^{(B{{\bf
u}^{t}})^{-}}$ for some nonzero ${\bf u}\in\mathbb{Z}^{2}$. Since $I_{A}$ is a
codimension 2 ideal and $C$ has minimal support, then there exists an
$i\not\in\operatorname{supp}(C)$ and ${\bf b}_{i}\not=(0,0)$. (Note that if
${\bf b}_{i}=(0,0)$, then ${\bf a}_{i}$ belongs to the free bouquet and
therefore $i$ does not belong to the support of any binomial in the form
$x^{{\bf v}^{+}}-x^{{\bf v}^{-}}$, where ${\bf
v}\in\operatorname{Ker}_{{\mathbb{Z}}}(A)$.) Then the matrix product ${\bf
b}_{i}{\mathbb{u}}^{t}=0$ implies ${\bf u}=\lambda{\widetilde{\bf b}_{i}}$ and
${\bf u},{\widetilde{\bf b}_{i}}$ are primitive vectors, they have the
greatest common divisor of their coordinates equal to 1, therefore $\lambda=1$
or $\lambda=-1$. Thus $C=x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf b}^{t}_{i}})}^{-}}$.

For the converse, suppose $C_{i}=x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf b}^{t}_{i}})}^{-}}$. Then for every
${\mathbb{a}}_{k}\in B_{i}$, the $i$-th bouquet of $A$, we have ${\bf
b}_{k}\cdot{\widetilde{\bf b}_{i}}=l\textbf{b}_{i}\cdot{\widetilde{\bf
b}_{i}}=0$, for some $l\in\mathbb{Q}$, therefore
$k\not\in\operatorname{supp}(C)$. Note also from the comments before Theorem
4.2 that for ${\mathbb{a}}_{i},{\mathbb{a}}_{j}$ in the same bouquet we have
either ${\widetilde{\bf b}_{i}}={\widetilde{\bf b}_{j}}$ or ${\widetilde{\bf
b}_{i}}=-{\widetilde{\bf b}_{j}}$. The complement of the support of
$C_{i}=x^{({B{\widetilde{\bf b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{-}}$ is the union of the elements of the bouquet that contains
${\mathbb{a}}_{i}$ and the elements of the free bouquet. Therefore, all the
$C_{i}=x^{({B{\widetilde{\bf b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{-}}$ are circuits. Furthermore, the cardinality of the set of
circuits is the number of non free bouquets.

######  Theorem 4.5.

Let $I_{A}$ be a codimension 2 toric ideal. Then $I_{A}$ is robust if and only
if it is strongly robust.

###### Proof.

Strongly robust toric ideals are always robust. Suppose that $I_{A}$ is
robust. From Lemma 4.4 $x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf b}^{t}_{i}})}^{-}}$ is a circuit and
circuits are always in the Universal Gröbner basis therefore
$x^{({B{\widetilde{\bf b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{-}}$ belongs to a minimal set of generators of $I_{A}$. By [26,
Theorem 5.10] every generator of a robust toric ideal is indispensable. Thus
also $x^{({B{\widetilde{\bf b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{-}}$ is indispensable, and therefore by Theorem 4.1
$-\widetilde{\bf b}_{i}$ belongs in $H_{1}\cup H_{2}\cup\cdots\cup
H_{n^{\prime}}$. Applying now [24, Lemma 2.7] we obtain that $I_{A}$ is
strongly robust.

In [5] Boocher et al. remarked on the central role that circuits play for the
robustness property of toric ideals of graphs. They asked the question: if the
circuits of $I_{A}$ are indispensable then is it true that the toric ideal is
robust? And they provide a counterexample of a toric ideal of a graph of
codimension 3 in which all circuits are indispensable but the ideal $I_{A}$ is
not robust. The graph $G_{t}$ in Fig.1 is a small alteration of their
counterexample, where in one of the two cut vertices, instead of one cycle of
length 3 we add $t$ cycles of length 3, $t\geq 1$. Every cycle of odd length
added on the second cut vertex increases the codimension by one. The toric
ideal of the graph with $t$ cycles has codimension $2+t$. The Graver basis, of
the toric ideal of $G_{t}$, is identical with the Universal Gröbner basis and
consists of $t(t-1)/2+8t+4$ elements in the form
$B_{1i}=e_{4}e_{7}e_{8}e_{12}^{2}c_{i}-e_{5}e_{6}e_{10}^{2}a_{i}b_{i}$,
$B_{2}=e_{4}e_{7}e_{9}e_{12}-e_{5}e_{6}e_{10}e_{11},$
$B_{3i}=e_{1}e_{5}^{2}e_{10}^{2}a_{i}b_{i}-e_{2}e_{3}e_{7}^{2}e_{12}^{2}c_{i}$,
$B_{4i}=e_{8}e_{11}e_{12}c_{i}-e_{9}e_{10}a_{i}b_{i},$
$B_{5}=e_{1}e_{4}e_{5}e_{8}-e_{2}e_{3}e_{6}e_{7}$,
$B_{6i}=e_{1}e_{4}^{2}e_{8}^{2}e_{12}^{2}c_{i}-e_{2}e_{3}e_{6}^{2}e_{10}^{2}a_{i}b_{i},$
$B_{7}=e_{1}e_{4}^{2}e_{8}e_{9}e_{12}-e_{2}e_{3}e_{6}^{2}e_{10}e_{11},$
$B_{8i}=e_{1}e_{4}^{2}e_{9}^{2}a_{i}b_{i}-e_{2}e_{3}e_{6}^{2}e_{11}^{2}c_{i},$
$B_{9}=e_{1}e_{5}^{2}e_{8}e_{10}e_{11}-e_{2}e_{3}e_{7}^{2}e_{9}e_{12},$
$B_{10i}=e_{4}e_{7}e_{9}^{2}a_{i}b_{i}-e_{5}e_{6}e_{8}e_{11}^{2}c_{i},$
$B_{11i}=e_{1}e_{5}^{2}e_{8}^{2}e_{11}^{2}c_{i}-e_{2}e_{3}e_{7}^{2}e_{9}^{2}a_{i}b_{i},$
$B_{12i}=e_{1}e_{4}e_{5}e_{9}e_{10}a_{i}b_{i}-e_{2}e_{3}e_{6}e_{7}e_{11}e_{12}c_{i},$
$B_{13(i,j)}=a_{i}b_{i}c_{j}-a_{j}b_{j}c_{i}$ where $i,j\in[t],$ see [21, 27].
All of them are circuits except $t$ of them in the form $B_{12i}$. The
circuits are minimal and indispensable since they do not have any chords. The
$t$ elements in the form $B_{12i}$ are not, since they are not minimal, every
one of them has an even chord, the $e_{8}$. For details see Theorems 4.13,
4.14 in [21] and 3.4 in [27].

Figure 1. The graph $G_{t}$

All these mean that for these toric ideals $I_{G_{t}}$ of codimension $2+t\geq
3$ all the circuits are indispensable and generate the toric ideal. The $t$
elements $B_{12i}$ are not minimal but they are elements of the Universal
Gröbner basis (and the Graver basis). We conclude that all $I_{G_{t}}$ are not
robust.

Therefore the answer to the question is negative for toric ideals of
codimension greater than three. For codimension 1 is obviously true since in
this case the ideal is principal. Next corollary shows that the answer to the
question of Boocher & al. ([5]) about circuits and robustness for codimension
two toric ideals is positive.

######  Corollary 4.6.

Let $I_{A}$ be a codimension 2 toric ideal. The circuits of $I_{A}$ are
indispensable if and only if $I_{A}$ is robust.

###### Proof.

It follows from Lemma 4.4 that the circuits are of the form
$x^{({B{\widetilde{\bf b}^{t}_{i}})}^{+}}-x^{({B{\widetilde{\bf
b}^{t}_{i}})}^{-}}$. Using Theorem 4.1 we obtain that the circuits of $I_{A}$
are indispensable if and only if both $\widetilde{\bf b}_{i}$ and
$-\widetilde{\bf b}_{i}$ are in $H_{1}\cup H_{2}\cup\cdots\cup H_{n}$ if and
only if $I_{A}$ is strongly robust, by [24, Lemma 2.7]. Finally, since $I_{A}$
is a co-dimension 2 toric ideal we have $I_{A}$ is strongly robust if and only
if it is robust, by Theorem 4.5.

The preceding discussions highlight the importance of better understanding the
properties of the strongly robustness simplicial complex $\Delta_{T}$ in all
codimensions. In particular, some of the key research directions to
investigate are the following:

  * •

Are there any classes of simple toric ideals $I_{T}$ for which we can
determine $\Delta_{T}$?

  * •

What kind of properties of $T$ determine $\Delta_{T}$?

  * •

When is $\Delta_{T}$ different than the empty simplicial complex?

  * •

What can one say about the dimension of $\Delta_{T}$?

Note that, in [19], Petrović et al. posed the question of whether every
strongly robust ideal has a mixed bouquet. In [24], Sullivant proved that
codimension 2 strongly robust toric ideals have at least two mixed bouquets
and posed the stronger question of whether every strongly robust ideal of
codimension $r$ has $r$ mixed bouquets. Finally, note that, in terms of the
strongly robustness simplicial complex $\Delta_{T}$, Sullivant’s question
becomes one about the dimension of $\Delta_{T}$. Namely, for every simple
codimension $r$ toric ideal $I_{T}$ in the polynomial ring of $s$ variables,
is it true that $\dim\Delta_{T}<s-r$?

Acknowledgments. The first author has been supported by the Royal Society
Dorothy Hodgkin Research Fellowship DHF$\backslash$R1$\backslash$201246\. The
third author has been partially supported by the grant PN-III-P4-ID-
PCE-2020-0029, within PNCDI III, financed by Romanian Ministry of Research and
Innovation, CNCS - UEFISCDI.

## References

  * [1] 4ti2 team, 4ti2 - a software package for algebraic, geometric and combinatorial problems on linear spaces, available at www.4ti2.de (2007).
  * [2] D. Bernstein, A. Zelevinsky, Combinatorics of maximal minors, J. Algebraic Combin. 2(2), 111–121 (1993).
  * [3] A. Boocher, Free resolutions and sparse determinantal ideals, Math. Res. Letters 19(4) 805–821 (2012).
  * [4] A. Boocher, E. Robeva, Robust toric ideals, J. Symbolic Comput., 68(1), 254–264 (2015).
  * [5] A. Boocher, B. C. Brown, T. Duff, L. Lyman, T. Murayama, A. Nesky and K. Schaefer, Robust graph ideals, Ann. Comb. 19(4), 641–660 (2015).
  * [6] M. Bourel, A. Dickenstein, A. Rittatore, Self-dual projective toric varieties, J. London Math. Soc. (2) 84, 514–540 (2011).
  * [7] H. Charalambous, A. Thoma, M. Vladoiu, Markov complexity of monomial curves, J. Algebra 417, 391–411 (2014).
  * [8] H. Charalambous, A. Thoma, M. Vladoiu, Markov bases and generalized Lawrence liftings, Ann. Comb. 19(4), 661–669 (2015).
  * [9] H. Charalambous, A. Thoma, M. Vladoiu, Binomial fibers and indispensable binomials, J. Symb. Comput. (74) 578–591 (2016).
  * [10] A. Conca, E. De Negri, E. Gorla, Universal Gröbner bases for maximal minors, International Mathematics Research Notices 11, 3245–3262 (2015).
  * [11] A. Conca, E. De Negri, E. Gorla, Cartwright–Sturmfels ideals associated to graphs and linear spaces. J. Comb. Algebra 2 (3), 231–257 (2018).
  * [12] P. Diaconis, B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Annals of Statistics 26(1), 363–397 (1998).
  * [13] I. Garcia-Marco, Ch. Tatakis, On robustness and related properties on Toric ideals, J. Alg. Comb. to appear.
  * [14] E. Gross, S. Petrović, Combinatorial degree bound for toric ideals of hypergraphs, Internat. J. Algebra Comput. 23 (6) 1503–1520 (2013).
  * [15] S. Hoşten, S. Sullivant, A finiteness theorem for Markov bases of hierarchical models, J. Combin. Theory Ser. A 114, 311–321 (2007).
  * [16] H. Ohsugi, T. Hibi, Toric ideals arising from contingency tables, Proceedings of the Ramanujan Mathematical Society’s Lecture Notes Series, 87–111, (2006).
  * [17] I. Ojeda, A. Vigneron-Tenorio, Indispensable binomials in semigroup ideals, Proc. Amer. Math. Soc. 138(12), 4205–4216 (2010).
  * [18] I. Peeva, B. Sturmfels, Syzygies of codimension 2 lattice ideals, Math. Z. 229, 163–194 (1998).
  * [19] S. Petrović, A. Thoma, M. Vladoiu, Bouquet algebra of toric ideals, J. Algebra 512, 493–525 (2018).
  * [20] S. Petrović, A. Thoma, M. Vladoiu, Hypergraph encodings of arbitrary toric ideals, J. Combin. Theory Ser. A 166, 11–41 (2019).
  * [21] E. Reyes, Ch. Tatakis, A. Thoma, Minimal generators of toric ideals of graphs, Adv. Appl. Math. 48, 64–78 (2012).
  * [22] B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, No 8, MAS, R.I. (1995).
  * [23] B. Sturmfels, A. Zelevinsky, Maximal minors and their leading terms, Adv. Math. 98(1), 65–112 (1993).
  * [24] S. Sullivant, Strongly Robust Toric Ideals in Codimension 2, J. Algebr. Stat. 10, 128–136 (2019).
  * [25] A. Takemura, S. Aoki, Distance-reducing Markov bases for sampling from a discrete sample space, Bernoulli, 11(5), 793–813 (2005).
  * [26] Ch. Tatakis, Generalized robust toric ideals, J. Pure Appl. Algebra 220, 263–277 (2016).
  * [27] Ch. Tatakis, A. Thoma, On the universal Gröbner bases of toric ideals of graphs, J. Combin. Theory Ser. A 118, 1540–1548 (2011).
  * [28] A. Thoma, M. Vladoiu, The defining matrices of self-dual projective toric varieties, in preparation.
  * [29] G. Ziegler, Lectures on Polytopes, Graduate Texts in Math. 152 Springer, New York (1995).