# Representations via differential algebras and equationally Noetherian
algebras

###### Abstract.

We show that free algebras of the variety of algebras generated by the Witt
algebra $W_{n}$, the left-symmetric Witt algebra $L_{n}$, and the symplectic
Poisson algebra $P_{n}$ can be described as subalgebras of differential
polynomial algebras with respect to appropriately defined products. Using
these representations, we prove that $W_{n}$, $L_{n}$, $P_{n}$, and the free
algebras of the varieties of algebras generated by these algebras are
equationally Noetherian.

Alexander A. Mikhalev111Faculty of Mechanics and Mathematics, M.V.Lomonosov
Moscow State University, Leninskie Gory 1, 119991, Moscow, Russian Federation
e-mail<EMAIL_ADDRESS>, Manat Mustafa222Department of Mathematics, SSH.,
Nazarbayev University, 53 Kabanbay Batyr Avenue, Astana, 010000, Kazakhstan,
e-mail<EMAIL_ADDRESS>and Ualbai Umirbaev333Department of
Mathematics, Wayne State University, Detroit, MI 48202, USA; and Institute of
Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan, e-mail:
<EMAIL_ADDRESS>

Mathematics Subject Classification (2020): Primary 12H05, 17B63, 17B66, 17D25;
Secondary 16P40, 17A50.

Key words: Differential algebras, Lie algebras, left-symmetric algebras,
Poisson algebras, equationally Noetherian algebras.

##  1\. Introduction

The well-known Hilbert Basis Theorem (see, for example [6]) implies that the
affine space $\mathbb{A}^{n}(K)$ over an arbitrary field $K$ is Noetherian. In
other words, any field $K$ is equationally Noetherian. The basics of algebraic
geometry over groups and algebraic systems can be found in [2, 4, 21]. Many
standard notions of algebraic geometry such as algebraic sets, the Zariski
topology, irreducible varieties, and coordinate groups were freely used to
solve the famous Tarski’s problems independently by O. Kharlampovich and A.
Myasnikov [11, 12] and by Z. Sela [25]. In 1986 V. Guba [8] proved that free
groups are equationally Noetherian using the well known representation of free
groups via matrices of order two. It is also known that linear groups over a
Noetherian domain [2] (in particular, polycyclic groups and finitely generated
metabelian groups), finitely generated free solvable groups [9], and torsion-
free hyperbolic groups [26] are equationally Noetherian.

There are only few examples of equationally Noetherian rings and algebras.
Obviously, Hilbert’s basis theorem implies that every associative and
commutative Noetherian ring or algebra is equationally Noetherian. It is also
known that any finitely generated metabelian or nilpotent Lie algebra is
equationally Noetherian [4]. Finitely generated free metabelian Lie algebras
of characteristic zero are Noetherian with respect to equations in the
universal enveloping algebras [24].

In this paper we give more examples of equationally Noetherian algebras. The
source of these examples is differential algebras. The Ritt-Raudenbush Basis
Theorem [10, 13, 23], which is an analogue of the Hilbert Basis Theorem, says
that every finitely generated differential algebra of characteristic zero is
radically Noetherian, i.e., satisfies the ascending chain condition for
radically closed differential ideals. Consequently, every finitely generated
differential domain over a field of characteristic zero is equationally
Noetherian.

We show that free algebras of the variety of algebras generated by the Witt
algebra $W_{n}$, the left-symmetric Witt algebra $L_{n}$, and the symplectic
Poisson algebra $P_{n}$ over a field of characteristic zero can be described
as subalgebras of differential polynomial algebras with respect to
appropriately defined products.

Using these representations, we prove that the algebras $W_{n}$, $L_{n}$,
$P_{n}$, and the free algebras of the varieties of algebras generated by these
algebras are equationally Noetherian.

Recall that the variety of algebras generated by the left symmetric Witt
algebra $L_{1}$ is the variety of all Novikov algebras [18]. Consequently,
every free Novikov algebra over a filed of characteristic zero is equationally
Noetherian. The solvability, nilpotency, and identities of Novikov algebras
have been intensively studied in recent years [27, 28, 30, 31]. In particular,
every Novikov algebra over a field of characteristic zero satisfying a
nontrivial polynomial identity is associative right nilpotent [5].

The paper is organized as follows. In Section 2 we give an exact definition of
equationally Noetherian algebras, formulate some basic results from [4], and
give some examples of equationally Noetherian free algebras. In Section 3 we
recall some well known results on differential algebras from [10, 13, 23] and
give some examples of equationally Noetherian differential algebras. Section 4
is devoted to a representation of free algebras of the variety of algebras
generated by the left-symmetric Witt algebra $L_{n}$ and the Witt algebra
$W_{n}$. Here we also prove that $W_{n}$, $L_{n}$, and the free algebras of
the varieties of algebras generated by these algebras are equationally
Noetherian. The same results for the symplectic Poisson algebra $P_{n}$ are
formulated in Section 5.

##  2\. On equationally Noetherian algebras

Let $\mathfrak{M}$ be an arbitrary variety of (universal) algebras and let $A$
be an arbitrary algebra of $\mathfrak{M}$. Let $\mathfrak{M}\langle
x_{1},x_{2},\ldots,x_{m}\rangle$ be the free algebra of the variety
$\mathfrak{M}$ freely generated by $x_{1},x_{2},\ldots,x_{m}$. Consider the
coproduct (or free product)

$\displaystyle B=A\sqcup_{\mathfrak{M}}\mathfrak{M}\langle
x_{1},x_{2},\ldots,x_{m}\rangle$

of $A$ and $\mathfrak{M}\langle x_{1},x_{2},\ldots,x_{m}\rangle$ in the
variety of algebras $\mathfrak{M}$. Recall that (see, for example [17]) there
exist homomorphisms $\alpha_{1}:A\to B$ and $\alpha_{2}:\mathfrak{M}\langle
x_{1},x_{2},\ldots,x_{m}\rangle\to B$ satisfying the following universal
property: for any algebra $C\in\mathfrak{M}$ and for any two homomorphisms
$\phi_{1}:A\to C$ and $\phi_{2}:\mathfrak{M}\to C$ there exists a unique
homomorphism $\phi:B\to C$ such that $\phi\circ\alpha_{1}=\phi_{1}$ and
$\phi\circ\alpha_{2}=\phi_{2}$. Notice that the homomorphisms $\alpha_{1}$ and
$\alpha_{2}$ are not always injections. In this paper we consider only the
varieties of algebras for which the homomorphisms $\alpha_{1}$ and
$\alpha_{2}$ are always injections. Then we can identify $A$ and
$\mathfrak{M}\langle x_{1},x_{2},\ldots,x_{m}\rangle$ with there images in
$B$. As a consequence of this, for any $m$-tuple $(a_{1},\ldots,a_{m})\in
A^{m}$ there exists the unique homomorphism $\phi:B\to A$ such that
$\phi_{A}=\mbox{id}$ and $\phi(x_{i})=a_{i}$ for all $i$. For any
$f=f(x_{1},\ldots,x_{m})\in B$ we set $f(a_{1},\ldots,a_{m})=\phi(f)$. This
allows to consider any element $f=f(x_{1},\ldots,x_{m})$ of $B$ as a function

$\displaystyle f:A^{m}\to A.$

For any subset $S$ of $B$ we define the set of all zeroes

$\displaystyle Z(S)=\\{(a_{1},\ldots,a_{m})|f(a_{1},\ldots,a_{m})=0,f\in S\\}$

of $S$ in $A^{m}$. The sets $Z(S)$ allow to define the Zariski topology on
$A^{m}$. The algebra $A$ is called equationally Noetherian if $A^{m}$ is
Noetherian with respect to Zariski topology for all natural $m$.

Notice that this definition does not depend on the choice of $\mathfrak{M}$
such that $A\in\mathfrak{M}$. In many cases it is more efficient to assume
that $\mathfrak{M}$ is the variety of algebras $\mathbf{Var}(A)$ generated by
$A$. Sometimes it is better to choose $\mathfrak{M}$ with some special
properties, for example, the varieties with an explicit description of the
coproduct of algebras. It might happen that $A\sqcup B=0$ for nonzero algebras
$A$ and $B$ in some varieties of algebras with unity in the signature.

######  Proposition 1.

[4] Let $A$ be an equationally Noetherian algebra. Then the following algebras
(of the same signature) are equationally Noetherian:

(1) Every subalgebra of $A$;

(2) Every direct power of $A$;

(3) Every algebra of the quasi variety of algebras $\mathbf{Qvar}(A)$
generated by $A$;

(4) Every free algebra of the variety of algebras $\mathbf{Var}(A)$ generated
by $A$.

Let us revise some examples of equationally Noetherian associative and
commutative algebras:

(i) Every field is equationally Noetherian. This is a corollary of Hilbert’s
basis theorem and is one of the basics of classical algebraic geometry.

(ii) Every domain is equationally Noetherian by Proposition 1(1) since it is
embeddable into a field.

(iii) Every Noetherian ring with identity is also equationally Noetherian by
Hilbert’s basis theorem.

(iv) Let $R$ be an arbitrary equationally Noetherian associative and
commutative algebra with identity containing the field of rationals
$\mathbb{Q}=\mathbb{Q}\cdot 1\subseteq R$. Then every polynomial algebra over
$R$ is equationally Noetherian. This follows from Proposition 1(4) since $R$
generates the variety of all associative and commutative algebras over $R$.

The next theorem gives more examples of associative and nonassociative
equationally Noetherian algebras.

######  Theorem 1.

Let $R$ be an arbitrary associative and commutative equationally Noetherian
ring with identity. Let $A$ be an arbitrary $R$-algebra that is a finitely
generated free module over $R$. Then $A$ is an equationally Noetherian
$R$-algebra.

Proof. For simplicity of notations, we prove this theorem only for rings $R$
containing the field of rational numbers $\mathbb{Q}$.

Let $e_{1},\ldots,e_{m}$ be a basis of a finite dimensional algebra $A$ over
$R$. Then every element $a\in A$ can be identified with
$(r_{1},\ldots,r_{m})\in R^{m}$ such that

$\displaystyle a=r_{1}e_{1}+\ldots+r_{m}e_{m},$

i.e.,

$\displaystyle A\equiv R^{m}$

and

(1) $\displaystyle A^{n}\equiv R^{mn}.$

Let $R_{0}=R[x_{i}^{j}:1\leq i\leq n,1\leq j\leq m]$ be a polynomial algebra
over $R$. Consider the algebra $A_{0}=R_{0}\otimes_{R}A$. Since $A_{0}$ is a
tensor product of free modules, $R_{0}$ and $A$ are embeddable into $A_{0}$.
Let $\mathfrak{M}=\mathrm{Var}(A)$ be the variety of algebras over $R$
generated by $A$. Since $\mathbb{Q}\subseteq R$, it follows that
$A_{0}\in\mathfrak{M}$.

Set

$\displaystyle X_{i}=x_{i}^{1}\otimes e_{1}+x_{i}^{2}\otimes
e_{2}+\ldots+x_{i}^{m}\otimes e_{m}\in A_{0},\ \ \ 1\leq i\leq n.$

Let $R\langle X_{1},\ldots,X_{n}\rangle$ be the $R$-subalgebra of $A_{0}$
generated by $X_{1},\ldots,X_{n}$. It is well known that $R\langle
X_{1},\ldots,X_{n}\rangle$ is the free algebra of the variety of algebras
$\mathfrak{M}=\mathrm{Var}(A)$ freely generated by $X_{1},\ldots,X_{n}$.

Let $C$ be the $R$-subalgebra of $A_{0}$ generated by $A$ and $R\langle
X_{1},\ldots,X_{n}\rangle$. Denote by $D=A\sqcup R\langle
X_{1},\ldots,X_{n}\rangle$ the coproduct of $A$ and $R\langle
X_{1},\ldots,X_{n}\rangle$ in $\mathfrak{M}$. Consider the homomorphism

$\displaystyle\omega:D\longrightarrow C$

of $R$-algebras that is identical on $A$ and $R\langle
X_{1},\ldots,X_{n}\rangle$.

For any homomorphism $\varphi:D\rightarrow A$ with $\varphi_{|A}=\mathrm{id}$
there exists the unique homomorphism $\varepsilon:C\rightarrow A$ such that
$\varepsilon_{|A}=\mathrm{id}$ and $\varphi=\varepsilon\circ\omega$. In fact,
if $\varphi(X_{i})=a_{i}\in A$ and

$\displaystyle a_{i}=\sum r_{i}^{j}\otimes e_{j},$

then we can define a homomorphism $\delta:A_{0}\to A$ that it is identical on
$A$ and $\delta(x_{i}^{j})=r_{i}^{j}$ for all $i,j$. We set
$\varepsilon=\delta_{|C}:C\to A$. Then $\varepsilon(X_{i})=a_{i}$ and
$\varepsilon$ satisfies all required conditions. The uniqueness of
$\varepsilon$ is obvious.

This means that if $f=f(X_{1},\ldots,X_{n})\in\mathrm{Ker}(\omega)$ then
$f(a_{1},\ldots,a_{n})=0$ for all $a_{1},\ldots,a_{n}\in A$. Consequently, the
kernel of $\omega$ is not essential in studying of equations over $A$.
Therefore it is sufficient to study equations of the form $s=0$ over $A$ where
$s\in C$.

Let $S$ be an arbitrary set of elements of $C$. Every element $s\in S$ can be
uniquely written in the form

$\displaystyle s=s_{1}\otimes e_{1}+\ldots+s_{m}\otimes e_{m}.$

Using the identification (1), the equation $s=0$ over $A$ is equivalent to the
equations $s_{1}=\ldots=s_{m}=0$ over $R$. Consequently, the set of equations
$s=0,s\in S,$ over $A$ is equivalent to the set of equations
$s_{1}=\ldots=s_{m}=0,s\in S$ over $R$. The latter is equivalent to a finite
subsystem of equations since $R$ is equationally Noetherian. Therefore the set
of equations $s=0,s\in S,$ over $A$ is also equivalent to a finite subsystem,
i.e., $A$ is equationally Noetherian. $\Box$

We formulate some essential corollaries of Theorem 1.

######  Corollary 1.

Every algebra of generic matrices over an equationally Noetherian associative
and commutative algebra is equationally Noetherian.

Proof. Let $R$ be an equationally Noetherian associative and commutative ring.
Then the algebra of matrices $M_{n}(R)$ of order $n$ over $R$ is equationally
Noetherian by Theorem 1. Recall that any algebra $G$ of generic matrices of
order $n$ is a free algebra of the variety of algebras
$\mathrm{Var}(M_{n}(R))$. Then $G$ is equationally Noetherian by Proposition
1. $\Box$

######  Corollary 2.

Every free algebra of the variety of associative algebras defined by the
identity

$\displaystyle[x_{1},y_{1}]\ldots[x_{n},y_{n}]=0$

over an equationally Noetherian associative and commutative algebra is
equationally Noetherian.

Proof. It is well known that this variety of associative algebras over $R$ is
generated by the algebra of triangular matrices $\mathrm{Tr}_{n}(R)$ [20]
since $\mathbb{Q}\subseteq R$. $\Box$

######  Corollary 3.

Every free $\mathfrak{N}_{c}\mathfrak{A}$-Lie algebra over an infinite field
is equationally Noetherian.

Proof. It is well known that this variety of Lie algebras is also generated by
the algebra of triangular matrices $\mathrm{Tr}_{n}(R)$ [1]. $\Box$

##  3\. Differential polynomial algebras

The basic concepts of differential algebras can be found in [10, 13, 23].

Let $\Delta=\\{\delta_{1},\ldots,\delta_{m}\\}$ be a basic set of derivation
operators.

A commutative ring $R$ with identity is said to be a differential ring or
$\Delta$-ring if all elements of $\Delta$ act on $R$ as a commuting set of
derivations, i.e., the derivations $\delta_{i}:R\rightarrow R$ are defined for
all $i$ and $\delta_{i}\delta_{j}=\delta_{j}\delta_{i}$ for all $i,j$.

Let $\Theta$ be the free commutative monoid on the set
$\Delta=\\{\delta_{1},\ldots,\delta_{m}\\}$ of derivation operators. The
elements

$\displaystyle\theta=\delta_{1}^{i_{1}}\ldots\delta_{m}^{i_{m}}$

of the monoid $\Theta$ are called derivative operators. The order of $\theta$
is defined as $|\theta|=i_{1}+\ldots+i_{m}$.

Let $R$ be a differential ring. Denote by $R^{e}$ the free left $R$-module
with a basis $\Theta$. Every element $u\in R^{e}$ can be uniquely written in
the form

$\displaystyle u=\sum_{\theta\in\Theta}r_{\theta}\theta$

with a finite number of nonzero $r_{\theta}\in R$. It is well known [15] that
there is the unique ring structure on $R^{e}$ defined by the relations

$\displaystyle\delta_{i}r=r\delta_{i}+\delta_{i}(r)$

for all $i$ and $r\in R$. Note that the ring $R^{e}$ is generated by $R$ and
$\Delta$. Every left module over $R^{e}$ is called a differential module over
$R$ [15]. For this reason we call $R^{e}$ the universal enveloping ring of
$R$.

Obviously, $R$ is a left $R^{e}$-module and every $I\subseteq R$ is a
differential ideal of $R$ if and only if $I$ is an $R^{e}$-submodule of $R$.

Let $x^{\Theta}=\\{x^{\theta}|\theta\in\Theta\\}$ be a set of symbols
enumerated by the elements of $\Theta$. Consider the polynomial algebra
$R[x^{\Theta}]$ over $R$ generated by the set of (polynomially) independent
variables $x^{\Theta}$. It is easy to check that the derivations $\delta_{i}$
can be uniquely extended to a derivation of $R[x^{\Theta}]$ by
$\delta_{i}(x^{\theta})=x^{\delta_{i}\theta}$. Denote this differential ring
by $R\\{x\\}$; it is called the ring of differential polynomials in $x$ over
$R$.

By adding additional variables, we can obtain the differential ring
$R\\{x_{1},x_{2},\ldots,x_{n}\\}$ of differential polynomials in
$x_{1},x_{2},\ldots,x_{n}$ over $R$. Let $M$ be the free commutative monoid
generated by all elements $x_{i}^{\theta}$, where $1\leq i\leq n$ and
$\theta\in\Theta$. The elements of $M$ are called differential monomials of
$R\\{x_{1},x_{2},\ldots,x_{n}\\}$. Every element $a\in
R\\{x_{1},x_{2},\ldots,x_{n}\\}$ can be uniquely written in the form

$\displaystyle a=\sum_{m\in M}r_{m}m$

with a finite number of nonzero $r_{m}\in R$.

An analog of the Hilbert Basis Theorem for differential algebras is known as
the Ritt-Raudenbush Basis Theorem [10]. The condition of maximality for
differential ideals does not hold for finitely generated differential rings.
The Ritt-Raudenbush Basis Theorem claims that if $R$ is a radically Notherian
ring containing the field of rational numbers $\mathbb{Q}$ then $R\\{x\\}$ is
also radically Noetherian, i.e., every radically closed differential ideal is
finitely generated.

Examples of equationally Noetherian differential rings are:

(i) Every differential field of characteristic zero is equationally
Noetherian. This is a direct corollary of the Ritt-Raudenbush Basis Theorem
and is one of the basics of differential algebraic geometry [13, Chapter IV].

(ii) Every differential domain of characteristic zero is equationally
Noetherian by Proposition 1(1) since it is embeddable into a differential
field.

In the remaining part of the paper we always assume that $R$ is an arbitrary
associative and commutative ring with identity containing the field of
rational numbers $\mathbb{Q}=\mathbb{Q}1\subseteq R$. Consider $R$ as a
$\Delta$-ring with $\delta_{i}(R)=0$ for all $i$. Then
$R\\{x_{1},x_{2},\ldots,x_{n}\\}$ becomes a $R$-algebra. Denote by
$C_{m}=R[x_{1},\ldots,x_{m}]$ the polynomial algebra over $R$ in the variables
$x_{1},\ldots,x_{m}$ considered as a differential algebra with derivations
$\delta_{i}\equiv\frac{\partial}{\partial x_{i}}$ for all $1\leq i\leq m$.

######  Proposition 2.

Let $C=R\\{y_{1},y_{2},...,y_{n},\ldots\\}$ be the differential polynomial
algebra over $R$ in the variables $y_{1},y_{2},...,y_{n},\ldots$. Let $f$ be a
nonzero element of $C$. Then there exists a homomorphism
$\varphi:C\longrightarrow C_{m}$ of differential $R$-algebras such that
$\varphi(f)\neq 0$.

Proof. Suppose that there exists a nonzero $f\in C$ such that $\varphi(f)=0$
for any homomorphism $\varphi:C\longrightarrow C_{m}$ of differential
$R$-algebras. Then $f=0$ is a nontrivial identity of the differential algebra
$C_{m}$. Since $\mathbb{Q}\subseteq R$, we can linearize the identities [32].
Linearizing $f$ we can get a nonzero polylinear element $g\in C$ such that
$g=0$ is an identity of $C_{m}$. Suppose that $g$ contains the variables
$y_{1},\ldots,y_{n}$ and is polylinear in these variables.

For any $\theta=\delta_{1}^{i_{1}}\ldots\delta_{m}^{i_{m}}\in\Theta$ we set
$\alpha(\theta)=(i_{1},\ldots,i_{m})\in\mathbb{Z}_{+}^{m}$ and

$\displaystyle X(\theta)=\frac{1}{i_{1}!\ldots i_{m}!}x_{1}^{i_{1}}\ldots
x_{m}^{i_{m}}\in C_{m}.$

If $\theta,\theta_{1}\in\Theta$ then we define set $\theta\leq\theta_{1}$ if
$\alpha(\theta)\leq\alpha(\theta_{1})$ with respect to the lexicographical
order in $\mathbb{Z}_{+}^{m}$. Notice that $X(\theta)^{\theta}=1$ and
$X(\theta)^{\theta_{1}}=0$ for all $\theta,\theta_{1}\in\Theta$ if
$\theta<\theta_{1}$.

For any polylinear differential monomial

$\displaystyle u=y_{1}^{\theta_{1}}y_{2}^{\theta_{2}}...y_{n}^{\theta_{n}}$

of degree $n$ set
$\beta(u)=(\alpha(\theta_{1}),\ldots,\alpha(\theta_{n}))\in\mathbb{Z}_{+}^{mn}$.
If $u$ and $v$ are two polylinear monomials then set $u\leq v$ if
$\beta(u)\leq\beta(v)$ with respect to the lexicographical order in
$\mathbb{Z}_{+}^{mn}$.

The polylinear element $g$ of degree $n$ can be uniquely written in the unique
form

$\displaystyle g=\alpha
y_{1}^{\theta_{1}}y_{2}^{\theta_{2}}...y_{n}^{\theta_{n}}+\sum\alpha_{i}m_{i},$

where $0\neq\alpha\in R$ and $m_{i}$ is a polylinear differential monomial
such that $y_{1}^{\theta_{1}}\ldots y_{n}^{\theta_{n}}<m_{i}$ for all $i$.

Notice that

$\displaystyle X(\theta_{1})^{\theta_{1}}\ldots X(\theta_{n})^{\theta_{n}})=1\
\mathrm{and}\ m_{i}(X(\theta_{1}),\ldots,X(\theta_{n}))=0$

for all $i$. Consequently,

$\displaystyle g(X(\theta_{1}),\ldots,X(\theta_{n}))=\alpha\neq 0,$

that is $g=0$ is not an identity of $C_{m}$. $\Box$

######  Corollary 4.

The variety of all differential algebras ($\Delta$-algebras) over $R$ is
generated by $C_{m}=R[x_{1},\ldots,x_{m}]$.

Proof. By Proposition 2, the algebra $C_{m}$ does not satisfy any nontrivial
differential identity. $\Box$

######  Theorem 2.

Let $A$ be a differential $R$-algebra without nilpotent elements satisfying
the ascending chain condition for radically closed differential ideals. Then
$A$ is equationally Noetherian.

Proof. Notice that $A\\{y_{1},\ldots,y_{n}\\}=A\sqcup
R\\{y_{1},\ldots,y_{n}\\}$ is the coproduct of $A$ and
$R\\{y_{1},\ldots,y_{n}\\}$ in the variety of differential $R$-algebras. Let
$S$ be any subset of $A\\{y_{1},\ldots,y_{n}\\}$ and

$\displaystyle Z(S)=\\{(a_{1},\ldots,a_{n})\in A^{n}|s(a_{1},\ldots,a_{n})=0\
\mathrm{for\ all}\ s\in S\\}.$

Let $I(S)$ be the set of all elements $f$ of $A\\{y_{1},\ldots,y_{n}\\}$ such
that $f(a_{1},\ldots,a_{n})=0$ for all $(a_{1},\ldots,a_{n})\in Z(S)$.
Obviously, $S\subseteq I(S)$ and $I(S)$ is an ideal of
$A\\{y_{1},\ldots,y_{n}\\}$. Note that

$\displaystyle
f(a_{1},\ldots,a_{n})^{\delta_{i}}=f^{\delta_{i}}(a_{1},\ldots,a_{n})$

for all $f\in A\\{y_{1},\ldots,y_{n}\\}$ and for all $a_{1},\ldots,a_{n}\in
A$. Consequently, $I(S)$ is a differential ideal of
$A\\{y_{1},\ldots,y_{n}\\}$.

If $f^{k}\in I(S)$ then $f^{k}(a_{1},\ldots,a_{n})=0$ for all
$(a_{1},\ldots,a_{n})\in Z(S)$. The algebra $R$ does not contain nilpotent
elements. Therefore $f(a_{1},\ldots,a_{n})=0$ and $f\in I(S)$. Thus $I(S)$ is
radically closed differential ideal of $A\\{y_{1},\ldots,y_{n}\\}$.

By the Ritt-Raudenbush basis Theorem [10, 13, 23], $A\\{y_{1},\ldots,y_{n}\\}$
satisfies the ascending chain condition for radically closed differential
ideals. Consequently, $I(S)$ coinsides with the radically closed differential
ideal generated by a finite number of elements $f_{1},\ldots,f_{k}\in
A\\{y_{1},\ldots,y_{n}\\}$. Therefore $Z(S)=Z(f_{1},\ldots,f_{k})$. $\Box$

######  Corollary 5.

If $R$ is an assiciative and commutative radically Noetherian domain then the
differential algebra $C_{m}=R[x_{1},\ldots,x_{m}]$ is equationally Noetherian.

Proof. Notice that the differential structure on $R$ is trivial, i.e., $R$ is
a radically Noetherian differential algebra. The algebra $C_{m}$ is radically
Noetherian differential $R$-algebra by Ritt-Raudenbuch Basis Theorem [10, 13,
23]. $\Box$

######  Corollary 6.

If $R$ is an associative and commutative radically Noetherian domain then
every differential polynomial $R$-algebra is equationally Noetherian.

Proof. Every free algebra of the variety of algebras $\mathrm{Var}(C_{m})$ is
equationally Noetherian by Proposition 1. $\Box$

##  4\. The left-symmetric Witt algebra $L_{m}$ and the Witt algebra $W_{m}$

Construction of left-symmetric and Lie algebras. Consider differential
algebras with the basic set of derivations
$\Delta=\\{\delta_{1},\ldots,\delta_{m}\\}$. Let $R$ be an arbitrary
associative and commutative ring with identity and $A$ be an arbitrary
differential algebra over $R$. Consider the space $B$ of elements of the form

$\displaystyle f_{1}\delta_{1}+f_{2}\delta_{2}+...+f_{n}\delta_{m},\ \
f_{i}\in A,1\leq i\leq m,$

of the universal enveloping algebra $C^{e}$. The product $\circ$ on $B$
defined by

$\displaystyle f\delta_{i}\circ g\delta_{j}=f\delta_{i}(g)\delta_{j},\ \ \
f,g\in C,1\leq i,j\leq m,$

turns $B$ into a left-symmetric algebra, that is $\langle B,\circ\rangle$
satisfies the identity

$\displaystyle(x\circ y)\circ z-x\circ(y\circ z)=(y\circ x)\circ
z-y\circ(x\circ z).$

This can be easily checked as in [16, 29]. We denote the left-symmetric
algebra $\langle B,\circ\rangle$ by $L(A)$ and call it the left-symmetric Witt
algebra of the differential algebra $A$.

The construction of $L(A)$ is a generalization of the well known Gelfand’s
construction for Novikov algebras [7], and if $m=1$ then $L(A)$ is a Novikov
algebra.

Recall that the variety of left-symmetric algebras is Lie admissible, i.e.,
the commutator algebra of any left-symmetric algebra is a Lie algebra.
Therefore $B$ is a Lie algebra with respect to the bracket

$\displaystyle[f\delta_{i},g\delta_{j}]=f\delta_{i}\circ
g\delta_{j}-g\delta_{j}\circ f\delta_{i}.$

Denote the Lie algebra $\langle B,[\cdot,\cdot]\rangle$ by $W(A)$ and call it
the Witt algebra of the differential algebra $A$. If $m=1$ then the bracket
$[\cdot,\cdot]$ is called the Wronskian [22].

The left-symmetric Witt algebra $L_{m}$ and the Witt algebra $W_{m}$. Let
$C_{m}=R[x_{1},x_{2},\ldots,x_{m}]$ be the differential algebra with
derivations $\delta_{i}=\partial_{i}$ for all $1\leq i\leq m$ on the
polynomial algebra $R[x_{1},x_{2},\ldots,x_{m}]$ . Then the left-symmetric
algebra $L_{m}=L(C_{m})$ is called the left-symmetric Witt algebra of index
$m$ and the Lie algebra $W_{m}=W(C_{m})$, is the well known Witt algebra of
index $m$ [16, 29].

Free algebras of the varieties $\mathrm{Var}(L_{m})$ and
$\mathrm{Var}(W_{m})$. Let $C=R\\{y_{ij}|1\leq i\leq n,1\leq j\leq m\\}$ be
the differential polynomial algebra in the variables $y_{ij}$ where $1\leq
i\leq n$ and $1\leq j\leq m$. We set

$\displaystyle Y_{i}=y_{i1}\delta_{1}+\ldots+y_{im}\delta_{m},\ \ \ 1\leq
i\leq n.$

Denote by ${\bf L}_{n}$ the subalgebra of the left-symmetric algebra $L(C)$
generated by $Y_{1},\ldots,Y_{n}$, and by ${\bf W}_{n}$ the subalgebra of the
Lie algebra $W(C)$ generated by $Y_{1},\ldots,Y_{n}$.

Below we show that ${\bf L}_{n}$ is the free algebra of the variety of
algebras $\mathrm{Var}(L_{m})$ freely generated by $Y_{1},\ldots,Y_{n}$ and
${\bf W}_{n}$ is the free algebra of the variety of algebras
$\mathrm{Var}(W_{m})$ freely generated by $Y_{1},\ldots,Y_{n}$.

”Coproducts”. Let $\widetilde{C}=C_{m}\\{y_{ij}|1\leq i\leq n,1\leq j\leq
m\\}$ be the differential polynomial algebra over the differential algebra
$C_{m}$ in the variables $y_{ij}$, where $1\leq i\leq n$ and $1\leq j\leq m$.
Obviously,

$\displaystyle\widetilde{C}=C_{m}\otimes_{R}C$

is the tensor product of the differential algebras $C_{m}$ and
$C=R\\{y_{ij}|1\leq i\leq n,1\leq j\leq m\\}$. Moreover, these algebras can be
identified with their images in $\widetilde{C}$ since they are free
$R$-modules. Then the left-symmetric algebra $L(\widetilde{C})$ contains
$L_{m}$ and ${\bf L}_{n}$, and the Lie algebra $W(\widetilde{C})$ contains
$W_{m}$ and ${\bf W}_{n}$.

Denote by $\widetilde{L}_{n}$ the subalgebra of the left-symmetric algebra
$L(\widetilde{C})$ generated by $L_{m}$ and ${\bf L}_{n}$ and by
$\widetilde{W}_{n}$ the subalgebra of the Lie algebra $W(\widetilde{C})$
generated by $W_{m}$ and ${\bf W}_{n}$.

######  Proposition 3.

$(a)$ For any $F_{1},\ldots,F_{n}\in L(\widetilde{C})$ there exists a unique
homomorphism $\phi:\widetilde{L}_{n}\to L(\widetilde{C})$ of the left-
symmetric algebras such that $\phi_{|L_{m}}=\mathrm{id}$ and
$\phi(Y_{i})=F_{i}$ for all $1\leq i\leq n$.

$(b)$ For any $F_{1},\ldots,F_{n}\in W(\widetilde{C})$ there exists a unique
homomorphism $\phi:\widetilde{W}_{n}\to W(\widetilde{C})$ of the Lie algebras
such that $\phi_{|W_{m}}=\mathrm{id}$ and $\phi(Y_{i})=F_{i}$ for all $1\leq
i\leq n$.

Proof. Consider

$\displaystyle F_{i}=f_{i1}\delta_{1}+\ldots+f_{im}\delta_{m},\ \
f_{ij}\in\widetilde{C},\ 1\leq i\leq n,\ 1\leq j\leq m.$

Obviously there exists a unique endomorphism
$\psi:\widetilde{C}\longrightarrow\widetilde{C}$ of the differential
$R$-algebra $\widetilde{C}$ such that $\psi_{|C_{m}}=\mathrm{id}$ and
$\psi(y_{ij})=f_{ij}$ for all $i,j$. This endomorphism can be uniquely
extended to an endomorphism of the enveloping algebra $\widetilde{C}^{e}$. We
denote this extension of $\psi$ by the same symbol, i.e.,
$\psi:\widetilde{C}^{e}\to\widetilde{C}^{e}$ is an endomorphism of the
associative algebra $\widetilde{C}^{e}$ such that $\psi(c)=c$ for all $c\in
C_{m}$, $\psi(y_{ij})=f_{ij}$, and $\psi(\delta_{j})=\delta_{j}$ for all
$i,j$. It is clear that $\psi(L(\widetilde{C}))\subseteq L(\widetilde{C})$ and

$\displaystyle\psi_{|\widetilde{L}_{n}}:\widetilde{L}_{n}\to L(\widetilde{C})$

is a homomorphism of the left-symmetric algebras. We set

$\displaystyle\phi=\psi_{|\widetilde{L}_{n}}:\widetilde{L}_{n}\to
L(\widetilde{C}).$

Then $\phi(Y_{i})=F_{i}$ for all $i$ and $\phi_{|L_{m}}=\mathrm{id}$. Such a
homomorphism is unique since $\widetilde{L}_{n}$ is generated by $L_{m}$ and
by $Y_{1},\ldots,Y_{n}$.

Repeating the same arguments we can prove the statement $(b)$ of the theorem.
$\Box$

######  Theorem 3.

$(a)$ The left-symmetric algebra ${\bf L}_{n}$ is the free algebra of the
variety of algebras $\mathrm{Var}(L_{m})$ freely generated by
$Y_{1},\ldots,Y_{n}$.

$(b)$ The Lie algebra ${\bf W}_{n}$ is the free algebra of the variety of
algebras $\mathrm{Var}(W_{m})$ freely generated by $Y_{1},\ldots,Y_{n}$.

Proof. Let $M$ be the free algebra of the variety of algebras
$\mathrm{Var}(L_{m})$ freely generated by $z_{1},\ldots,z_{n}$. It is well
known that $M$ is isomorphic to a subalgebra of a direct power of $L_{m}$ [3,
19].

Let $I$ be an arbitrary set of indexes, $L_{m}^{I}$ be the direct $I$-power of
$L_{m}$, and assume that $M$ is isomorphic to a subalgebra of $L_{m}^{I}$. Let
$F_{1},\ldots,F_{n}$ be any set of elements of $L_{m}^{I}$. Suppose that
$F_{i}=(f_{ij})_{j\in I}$. By Proposition 3 for each $j\in I$ there exists a
homomorphism $\psi_{j}:\widetilde{L}_{n}\to L_{m}$ such that
$\phi_{j}(Y_{i})=f_{ij}$ for all $i$. For $\phi_{j}={\phi_{j}}_{|{{\bf
L}_{n}}}$ we have that

$\displaystyle\phi:{\bf L}_{n}\to L_{m}^{I},$

defined by $\phi(a)=(\phi_{j}(a))_{j\in I}$ for all $a\in{\bf L}_{n}$, is a
homomorphism of the left symmetric algebras such that $\phi(Y_{i})=F_{i}$ for
all $i$.

It follows that there exists a homomorphism

$\displaystyle\tau:{\bf L}_{n}\to M,$

such that $\tau(Y_{i})=z_{i}$ for all $1\leq i\leq n$.

If $\tau$ is not an isomorphism then there exists a nonzero element
$F=F(Y_{1},\ldots,Y_{n})$ such that $\phi(F)=0$ for all homomorphisms
$\phi:{\bf L}_{n}\to L_{m}$. Consider

$\displaystyle F=f_{1}\delta_{1}+\ldots+f_{m}\delta_{m},f_{i}\in C.$

Suppose that $f_{i}\neq 0$. By Proposition 2, there exists a homomorphism
$\psi:C\to C_{m}$ such that $\psi(f_{i})\neq 0$. As above, we can extend this
homomorphism to $\psi:C^{e}\to C_{m}^{e}$. If $\phi=\psi_{|L}:L\to L_{m}$,
then

$\displaystyle\phi(F)=\psi(f_{1})\delta_{1}+\ldots+\psi(f_{i})\delta_{i}+\ldots+\psi(f_{m})\delta_{m}\neq
0$

since $\psi(f_{i})\neq 0$. This contradiction proves that the kernel of $\tau$
is $0$, i.e., $\tau$ is an isomorphism.

This proves the statement $(a)$ of the theorem. The statement $(b)$ can be
proved similarly. $\Box$

######  Theorem 4.

Let $R$ be an arbitrary radically Noetherian domain containing the field of
rational numbers $\mathbb{Q}=\mathbb{Q}1\subseteq R$. Then

$(a)$ the left-symmetric Witt algebra $L_{m}$ over $R$ is equationally
Noetherian;

$(b)$ the Witt algebra $W_{m}$ over $R$ is equationally Noetherian.

Proof. By Proposition 3, for any natural $n\geq 0$ and for any system of
elements $F_{1},\ldots,F_{n}\in L(\widetilde{C})$ there exists a homomorphism
$\phi:{\bf L}_{n}\to L(\widetilde{C})$ of the left-symmetric algebras such
that $\phi(Y_{i})=F_{i}$ for all $1\leq i\leq n$. Therefore $L(\widetilde{C})$
belongs to the variety of algebras $\mathrm{Var}(L_{m})$.

Let $L_{m}\sqcup{\bf L}_{n}$ be the coproduct of $L_{m}$ and ${\bf L}_{n}$ in
the variety of algebras $\mathrm{Var}(L_{m})$. Since $\widetilde{L}_{n}$ is
the subalgebra of $L(\widetilde{C})$ generated by $L_{m}$ and ${\bf L}_{n}$,
there exists a unique homomorphism

$\displaystyle\phi:L_{m}\sqcup{\bf L}_{n}\to\widetilde{L}_{n}$

such that $\phi_{|L_{m}}=\mathrm{id}$ and $\phi_{|{\bf L}_{n}}=\mathrm{id}$.

Let $S$ be any subset of $L_{m}\sqcup{\bf L}_{n}$ and

$\displaystyle Z(S)=\\{(a_{1},\ldots,a_{n})\in
L_{m}^{n}|s(a_{1},\ldots,a_{n})=0\ \mathrm{for\ all}\ s\in S\\}.$

Recall that $s(a_{1},\ldots,a_{n})$ is the image of $s$ under the homomorphism
$L_{m}\sqcup{\bf L}_{n}\to L_{m}$ that is identical on $L_{m}$ and sends each
$Y_{i}$ to $a_{i}$ for all $i$. By Proposition 3, $s(a_{1},\ldots,a_{n})=0$ if
and only if $\phi(s)(a_{1},\ldots,a_{n})=0$. Therefore

$\displaystyle Z(S)=\\{(a_{1},\ldots,a_{n})\in
L_{m}^{n}|\phi(s)(a_{1},\ldots,a_{n})=0\ \mathrm{for\ all}\ s\in S\\}.$

Suppose that

$\displaystyle\phi(s)=s_{1}\delta_{1}+\ldots+s_{m}\delta_{m},\
s_{1},\ldots,s_{m}\in\widetilde{C},$

and

$\displaystyle a_{i}=a_{i1}\delta_{1}+\ldots+a_{im}\delta_{m},\
a_{i1},\ldots,a_{im}\in C_{m},\ 1\leq i\leq n.$

Then

$\displaystyle\phi(s)(a_{1},\ldots,a_{n})=\sum_{k=1}^{m}s_{k}(a_{ij})\delta_{k}.$

Let $I(S)$ be the set of all elements $f$ of $\widetilde{C}$ such that
$f(a_{1},\ldots,a_{n})=0$ for all $(a_{1},\ldots,a_{n})\in Z(S)$. Obviously,
$s_{i}\in I(S)$ for all $s\in S$ and $1\leq i\leq m$. Moreover, $I(S)$ is an
ideal of $\widetilde{C}$. We have

$\displaystyle f(a_{ij})^{\delta_{k}}=f^{\delta_{k}}(a_{ij})$

for all $f\in\widetilde{C}$ and for all $a_{ij}\in C_{m}$. It follows that
$I(S)$ is a differential ideal of $\widetilde{C}$.

If $f^{k}\in I(S)$ then $f^{k}(a_{ij})=0$ for all $(a_{1},\ldots,a_{n})\in
Z(S)$. Since $R$ is a domain, the algebras $C_{m}=R[x_{1},\ldots,x_{m}]$ and
$\widetilde{C}=C_{m}\\{y_{ij}|1\leq i\leq n,1\leq j\leq m\\}$ do not contain
nilpotent elements. Therefore $f(a_{ij})=0$ and $f\in I(S)$. Thus $I(S)$ is
radically closed differential ideal of $\widetilde{C}$.

By the Ritt-Raudenbush Basis Theorem [10, 13, 23], $C_{m}$ and $\widetilde{C}$
satisfy the ascending chain condition for radically closed differential
ideals. Therefore there exists a finite subset $T$ of $S$ such that the
radically closed differential ideal of $\widetilde{C}$ generated by all
$t_{i},1\leq i\leq m,t\in T$ coinsides with $I(S)$. It implies that
$Z(S)=Z(T)$.

This proves the statement $(a)$ of the theorem. The statement $(b)$ can be
proved similarly. $\Box$

######  Corollary 7.

Let $R$ be an arbitrary radically Noetherian domain containing the field of
rational numbers $\mathbb{Q}=\mathbb{Q}1\subseteq R$. Then

$(a)$ every free algebra over $R$ of the variety of $\mathrm{Var}(L_{m})$ is
equationally Noetherian;

$(b)$ every free algebra over $R$ of the variety of $\mathrm{Var}(W_{m})$ is
equationally Noetherian.

Proof. This is a direct corollary of Proposition 1 and Theorem 4. $\Box$

##  5\. Symplectic Poisson algebra $P_{m}$

Construction of Poisson algebras. Consider differential algebras with the
basic set of derivations $\Delta=\\{\delta_{1},\ldots,\delta_{2m}\\}$
containing $2m$ elements. Let $R$ be an arbitrary associative and commutative
ring with identity and let $A$ be an arbitrary differential algebra over $R$.
For any $f,g\in A$ we set

$\displaystyle\\{f,g\\}=\sum_{i=1}^{m}(\delta_{2i-1}(f)\delta_{2i}(g)-\delta_{2i}(f)\delta_{2i-1}(g)).$

It is easy to show that the $R$-module $A$ with respect to the product $\cdot$
and the bracket $\\{\cdot,\cdot\\}$ is a Poisson algebra [14], in other words,

$(a)$ $\langle A,\cdot\rangle$ is an associative and commutative algebra,

$(b)$ $\langle A,\\{\cdot,\cdot\\}\rangle$ is a Lie algebra, and

$(c)$ $\langle A,\cdot,\\{\cdot,\cdot\\}\rangle$ satisfies the (Leibniz)
identity:

$\displaystyle\\{xy,z\\}=x\\{y,z\\}+\\{x,y\\}y.$

Denote the Poisson algebra $\langle A,\cdot,\\{\cdot,\cdot\\}\rangle$ by
$P(A)$.

The symplectic Poisson algebra $P_{m}$. Let $C_{2m}=R[x_{1},\ldots,x_{2m}]$ be
the differential algebra with derivations $\delta_{i}=\partial_{i}$ for all
$1\leq i\leq 2m$ on the polynomial algebra $R[x_{1},\ldots,x_{2m}]$. We denote
$P_{m}=P(C_{2m})$. The Poisson algebra $P_{m}$ is called the symplectic
Poisson algebra of index $m$ over $R$. If $f,g\in P_{m}$ then

$\displaystyle\\{f,g\\}=\sum_{i=1}^{m}(\frac{\partial f}{\partial
x_{2i-1}}\frac{\partial g}{\partial x_{2i}}-\frac{\partial f}{\partial
x_{2i}}\frac{\partial g}{\partial x_{2i-1}}).$

Free algebras of the variety $\mathrm{Var}(P_{m})$. Let
$C=R\\{y_{1},\ldots,y_{n}\\}$ be the differential polynomial algebra over $R$
in the variables $y_{1},\ldots,y_{n}$. Denote by ${\bf P}_{n}$ the subalgebra
of the Poisson algebra $P(C)$ generated by $y_{1},\ldots,y_{n}$.

Below we show that ${\bf P}_{n}$ is the free algebra of the variety of
algebras $\mathrm{Var}(P_{m})$ freely generated by $y_{1},\ldots,y_{n}$.

”Coproducts”. Let $\widetilde{C}=C_{2m}\\{y_{1},\ldots,y_{n}\\}$ be the
differential polynomial algebra over the differential algebra $C_{2m}$ in the
variables $y_{1},\ldots,y_{n}$. It is clear that

$\displaystyle\widetilde{C}=C_{2m}\otimes_{R}C$

is the tensor product of the differential algebras $C_{2m}$ and
$C=R\\{y_{1},\ldots,y_{n}\\}$. Moreover, these algebras can be identified with
their images in $\widetilde{C}$ since they are free $R$-modules. Then the
Poisson algebra $P(\widetilde{C})$ contains $P_{m}$ and ${\bf P}_{n}$.

Denote by $\widetilde{P}_{n}$ the subalgebra of the Poisson algebra
$P(\widetilde{C})$ generated by $P_{m}$ and ${\bf P}_{n}$.

######  Proposition 4.

For any $F_{1},\ldots,F_{n}\in P(\widetilde{C})$ there exists the unique
homomorphism $\phi:\widetilde{P}_{n}\to P(\widetilde{C})$ of the Poisson
algebras such that $\phi_{|P_{m}}=\mathrm{id}$ and $\phi(y_{i})=F_{i}$ for all
$1\leq i\leq n$.

The proof of this proposition repeats the proof of Proposition 3. The proofs
of the following two theorems are similar to the proofs of Theorem 3 and
Theorem 4, respectively.

######  Theorem 5.

The Poisson algebra ${\bf P}_{n}$ is the free algebra of the variety of
algebras $\mathrm{Var}(P_{m})$ freely generated by $y_{1},\ldots,y_{n}$.

######  Theorem 6.

Let $R$ be an arbitrary radically Noetherian domain containing the field of
rational numbers $\mathbb{Q}=\mathbb{Q}1\subseteq R$. Then the Poisson algebra
$P_{m}$ over $R$ is equationally Noetherian.

######  Corollary 8.

Let $R$ be an arbitrary radically Noetherian domain containing the field of
rational numbers $\mathbb{Q}=\mathbb{Q}1\subseteq R$. Then every free algebra
over $R$ of the variety $\mathrm{Var}(P_{m})$ is equationally Noetherian.

Using Proposition 3, Theorem 3, and general results on the structure of the
variety of algebras generated by an algebra [3, 19], it is possible to show
that $\widetilde{L}_{n}$ is the coproduct of $L_{m}$ and ${\bf L}_{n}$ in the
variety of algebras $\mathrm{Var}(L_{m})$ and $\widetilde{W}_{n}$ is the
coproduct of $W_{m}$ and ${\bf W}_{n}$ in the variety of algebras
$\mathrm{Var}(W_{m})$. Similarly it can be proved that $\widetilde{P}_{n}$ is
the coproduct of $P_{m}$ and ${\bf P}_{n}$ in the variety of algebras
$\mathrm{Var}(P_{m})$.

Acknowledgments

The first author is supported by Russian Science Foundation, grant
22-11-00052.

The second author is supported by Nazarbayev University FDCRG (Grant No.
021220FD3851).

The third author is supported by the Ministry of Education and Science of the
Republic of Kazakhstan (project AP14872073).

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# Toric Ideals of Weighted Oriented Graphs

Jennifer Biermann  Department of Mathematics and Computer Science, Hobart and
William Smith Colleges
300 Pulteney St. Geneva, NY 14456, USA<EMAIL_ADDRESS>,  Selvi Kara
Department of Mathematics, University of Utah, 155 1400 E, Salt Lake City, UT
84112, USA<EMAIL_ADDRESS>,  Kuei-Nuan Lin  Department of Mathematics,
Penn State University Greater Allegheny
4000 University Dr, McKeesport, PA 15132, USA<EMAIL_ADDRESS>and  Augustine
O’Keefe  Department of Mathematics and Statistics, Connecticut College
270 Mohegan Avenue Pkwy, New London, CT 06320, USA<EMAIL_ADDRESS>
###### Abstract.

Given a vertex-weighted oriented graph, we can associate to it a set of
monomials. We consider the toric ideal whose defining map is given by these
monomials. We find a generating set for the toric ideal for certain classes of
graphs which depends on the combinatorial structure and weights of the graph.
We provide a result analogous to the unweighted, unoriented graph case, to
show that when the associated simple graph has only trivial even closed walks,
the toric ideal is the zero ideal. Moreover, we give necessary and sufficient
conditions for the toric ideal of a weighted oriented graph to be generated by
a single binomial and we describe the binomial in terms of the structure of
the graph.

2020 Mathematics Subject Classification. Primary 13F65, 13A70; Secondary
05C50, 05C38, 05E40, 13C05, 13F55, 05C20, 14M25

##  1\. Introduction

Toric ideals are of general interest due to a multitude of applications to
other fields. One such application is using toric methods for analyzing
chemical reaction networks, [4, 6]. Given a connected chemical reaction
network one can associate to it a monomial ideal which, in turn, gives rise to
a toric ideal called the complex balancing ideal of the toric dynamical
system. See [3] for a more recent treatment from the commutative algebraic
viewpoint of multi-Rees algebras.

Within the field of algebraic statistics, toric ideals arise as the vanishing
ideal of discrete exponential families, also known as toric models. Generating
sets of the toric ideals give information about Markov chains on sets of
contingency tables [5]. As such, a generating set of the toric ideal is called
a Markov basis. It is also known that the degree of the toric ideal gives rise
to an upper bound for the maximum likelihood degree of the associated toric
model [2].

Toric ideals are defined via the kernel of a monomial map. When the monomials
defining the map are square-free, one can use the methods of toric ideals of
(hyper)graphs [11, 12, 15]. In general, however, the defining monomials of the
toric ideals in these applications are not square-free. Recently, it has been
shown that the combinatorial structure of general toric ideals can be encoded
by those arising from square-free monomials [13]. However, the fundamental
problem of finding an explicit generating set for toric ideals defined by non-
square-free monomials is still open.

In this paper, we consider toric ideals arising from the edge ideals of
vertex-weighted oriented graphs (VWOGs) which themselves have applications in
coding theory, specifically in Reed-Muller-type codes [10]. The study of
ideals associated to VWOGs is very recent, and consequently, we have few
results in this direction. In the case of monomial edge ideals of VWOGs,
Cohen-Macaulayness has been studied [7, 9, 14], and the authors calculated the
regularity of classes of edge ideals of VWOGs [1].

The goal of this work is to characterize a generating set for the ideal based
on the combinatorial structure of the graph. In the case of unoriented simple
graphs, a generating set of the associated toric ideal is given by the even
closed walks in the graph [11, 15]. In the case of VWOGs, not every even
closed walk gives an element of the toric ideal. Additionally, we give
examples where the toric ideal of a VWOG has minimal generators corresponding
to odd walks in the graph which would not happen in the unoriented case (see
Example 5.5 and Example 6.2). So we see that the situation for the toric
ideals of VWOGs is much more complicated than that of the unoriented case.

The remainder of this paper is structured as follows. We first prove a result,
Theorem 3.6, that is analogous to the unoriented, unweighted case that shows
that if the radical of a monomial ideal is associated to a simple graph that
has only trivial even closed walks then the toric ideal is the zero ideal. The
simplest non-zero toric ideals of weighted oriented graphs are therefore even
cycles. We give a combinatorial description (which we call _balanced_ , see
Definition 4.2) of when a weighted oriented even cycle has a non-zero toric
ideal in (Theorem 4.3). Unlike the edge ideal case, a weighted oriented graph
that consists of two cycles connected via a vertex or a path always has a non-
zero toric ideal regardless of the parity of the lengths of the cycles. In
Theorem 6.1, building off of the results on VWOGs with two cycles in Section
5, we characterize when the toric ideal of a VWOG has a single generator.

Our main results are summarized in the following.

######  Theorem 1.1.

  1. (1)Let $M$ be a monomial ideal such that $\sqrt{M}$ is generated by square-free monomials of degree 2 and its associated graph has only trivial even closed walks, then the toric ideal associated to $M$ is a zero ideal.

  2. (2)

Let $\mathcal{D}$ be a weighted oriented cycle on $n$ vertices. The toric
ideal associated to $\mathcal{D}$, $I_{\mathcal{D}}$, is non-zero if and only
if $\mathcal{D}$ is balanced.

  3. (3)

Let $\mathcal{D}$ be a weighted oriented graph. We provide necessary and
sufficient conditions for which associated toric ideal $I_{\mathcal{D}}$ is
generated by a single element.

##  2\. Preliminaries

###  2.1. Toric Ideals and Rings

Let $M$ be a monomial ideal in polynomial ring $k[x_{1},\dots,x_{\nu}]$ with
minimal generating set $\operatorname{gens}(M)=\\{\mathbf{x}^{{\bf
a}_{1}},\mathbf{x}^{{\bf a}_{2}},\dots,\mathbf{x}^{{\bf a}_{\mu}}\\}$, and
$k[e_{1},\dots,e_{\mu}]$ a polynomial ring with as many indeterminates as
elements of $\operatorname{gens}(M)$. One can then define a $k$-algebra
homomorphism $\varphi:k[e_{1},\dots,e_{\mu}]\rightarrow
k[x_{1},\dots,x_{\nu}]$ via the monomial map $e_{i}\mapsto\mathbf{x}^{{\bf
a}_{i}}$. The kernel of $\varphi$ is the toric ideal of $M$, denoted $I_{M}$,
and the toric ring of $M$ is the image of $\varphi$, denoted $k[M]$. Note that
$k[M]$ has a natural multigrading by $\mathbb{N}\\{{\bf a}_{1},{\bf
a}_{2},\dots,{\bf a}_{\mu}\\}$. Let ${\bf
a}_{j}=(a_{1,j},\dots,a_{\nu,j})\in\mathbb{N}^{\nu}$, and let $A(M)=(a_{i,j})$
be the $\nu\times\mu$ incidence matrix of $M$ with $\mathscr{A}=\\{{\bf
a}_{1},\dots,{\bf a}_{\mu}\\}$ the set of columns in $A(M)$. It is well known
that $I_{\mathcal{D}}$ is generated by irreducible binomials
$\prod_{i=1}^{\mu}e_{i}^{r_{i}}-\prod_{i=1}^{\mu}e_{i}^{s_{i}}$ where the
monomials $\prod_{i=1}^{\mu}e_{i}^{r_{i}}$ and
$\prod_{i=1}^{\mu}e_{i}^{s_{i}}$ have the same multidegree in
$\mathbb{N}\mathscr{A}$, i.e. $\sum_{i=1}^{\mu}r_{i}{\bf
a}_{i}=\sum_{i=1}^{\mu}s_{i}{\bf a}_{i}$ and $r_{i}$ and $s_{i}$ cannot both
be non-zero for any $i$. For a particular element
$f=\prod_{i=1}^{\mu}e_{i}^{r_{i}}-\prod_{i=1}^{\mu}e_{i}^{s_{i}}$ we say the
support of $f$, denoted $\operatorname{supp}f$, is the set of variables
$e_{i}$ which occur in $f$ with non-zero exponent.

###  2.2. Edge Ideals of Graphs and Weighted Oriented Graphs

Let $G=(V(G),E(G))$ be a simple graph with vertex set
$V(G)=\\{x_{1},\dots,x_{\nu}\\}$ and edge set
$E(G)=\\{e_{1}=\\{x_{1_{1}},x_{1_{2}}\\},\dots,e_{\mu}=\\{x_{\mu_{1}},x_{\mu_{2}}\\}\\}$.
We adopt the definitions of cycle, tree, induced subgraph, leaf, walk, closed
walk, and trivial closed walk from the literature of simple graphs. We recall
some of the definitions here to be used in this work’s content.

######  Definition 2.1.

A _walk of length $q$_ of a graph $G$ connecting $x_{i_{1}}\in V(G)$ and
$x_{i_{q+1}}\in V(G)$ is a finite sequence of the form

$W=(\\{x_{i_{1}},x_{i_{2}}\\},\\{x_{i_{2}},x_{i_{3}}\\},\dots,\\{x_{i_{q}},x_{i_{q+1}}\\})$

with each $\\{x_{i_{k}},x_{i_{k+1}}\\}\in E(G)$. An _even (resp. odd) walk_ is
a walk of even (resp. odd) length. A walk $W$ is called _closed_ if
$x_{i_{1}}=x_{i_{q+1}}$, i.e. the initial vertex is the same as the ending
vertex. We say $W$ is a _path_ if $x_{i_{j}}\neq x_{i_{k}}$ for all $1\leq
j<k\leq q$. A _cycle_ is a closed path. A _trivial closed walk_ is a closed
walk having no induced cycles.

Let $M=(x_{i_{1}}x_{i_{2}}|\\{x_{i_{1}},x_{i_{2}}\\}\in E(G))$ be a square-
free monomial ideal associated to the graph $G$. We call $M$, the _edge ideal
of_ $G$. Conversely, given a monomial ideal, $M$, such that
$\operatorname{gens}(M)$ is a set of degree 2 square-free monomials,
$x_{1_{1}}x_{1_{2}},\dots,x_{{\mu}_{1}}x_{\mu_{2}}\in k[x_{1},\dots,x_{\nu}]$,
then one can construct a simple graph $G_{M}$ with vertex set
$\\{x_{1},\dots,x_{\nu}\\}$ and edge set
$\\{\\{x_{1_{1}},x_{1_{2}}\\},\dots,\\{x_{\mu_{1}},x_{\mu_{2}}\\}\\}$. The
following is the well-known characterization of the generators of $I_{M}$
found in [11] and [15].

######  Theorem 2.2.

When $M$ is the square-free monomial ideal associated to a simple graph $G$,
the toric ideal $I_{M}$ is generated by even closed walks of $G$.

######  Example 2.3.

Let
$\operatorname{gens}(M)=\\{x_{1}x_{2},x_{2}x_{3},x_{3}x_{1},x_{1}x_{4},x_{4}x_{5},x_{5}x_{1}\\}$,
then $G_{M}$ is comprised of two 3-cycles joined at a vertex with
$E(G)=\\{e_{1}=\\{x_{1},x_{2}\\},e_{2}=\\{x_{2},x_{3}\\},\dots,e_{6}=\\{x_{5}x_{1}\\}\\}$.
The shortest nontrivial even closed walk in $G_{M}$ traverses each edge of the
graph once, and $I_{M}=(e_{1}e_{3}e_{5}-e_{2}e_{4}e_{6})$.

######  Remark 2.4.

Theorem 2.2 shows that if the graph of a square-free monomial ideal, $M$, has
only trivial even closed walks, then the toric ideal $I_{M}$ is the zero
ideal.

Let $\mathcal{D}=(V(\mathcal{D}),E(\mathcal{D}))$ be a weighted oriented graph
with weight vector ${\bf w}\in\mathbb{N}^{|V(\mathcal{D})|}$ where
$V(\mathcal{D})=\\{x_{1},\dots,x_{\nu}\\}$,
$E(\mathcal{D})=\\{e_{1},\dots,e_{\mu}\\}$, and ${\bf
w}=(w_{1},w_{2},\dots,w_{\nu})$. We abuse notation and define the polynomial
rings $k[E(\mathcal{D})]=k[e_{1},\dots,e_{\mu}]$ and
$k[V(\mathcal{D})]=k[x_{1},\dots,x_{\nu}]$. We then consider the $k$-algebra
homomorphism $\varphi:k[e_{1},\dots,e_{\mu}]\rightarrow
k[x_{1},\dots,x_{\nu}]$ defined by
$\varphi(e_{i})=x_{i_{1}}x_{i_{2}}^{w_{i_{2}}}$ where $e_{i}$ is the directed
edge from $x_{i_{1}}$ to $x_{i_{2}}$.

######  Notation 2.5.

We use the ordered pair $(x_{i_{1}},x_{i_{2}})$ to denote the oriented edge
from $x_{i_{1}}$ to $x_{i_{2}}$. We write $e=\\{x_{i_{1}},x_{i_{2}}\\}$ when
the oriented edge $e$ is incident to $x_{i_{1}}$ and $x_{i_{2}}$ but the
orientation is unspecified.

Let $A(\mathcal{D})=(a_{i,j})$ be the $\nu\times\mu$ incidence matrix of
$\mathcal{D}$ defined by

$a_{i,j}=\begin{cases}1&\text{ if $e_{j}=(x_{i},x_{k})\in E(\mathcal{D})$ for
some $1\leq k\leq\nu$},\\\ w_{i}&\text{ if $e_{j}=(x_{k},x_{i})\in
E(\mathcal{D})$ for some $1\leq k\leq\nu$},\\\ 0&\text{ otherwise}\end{cases}$

with $\mathscr{A}=\\{{\bf a}_{1},\dots,{\bf a}_{\mu}\\}$ the set of columns in
$A(\mathcal{D})$.

Note that the monomials defining the $k$-algebra homomorphism $\varphi$ are
exactly the generators of the monomial edge ideal studied in [1, 7, 9, 10,
14]. Defining the toric edge ring of $\mathcal{D}$ as the image of the
monomial map defined by the generators of the monomial edge ideal is a natural
analogue to defining the toric edge ring of an unoriented finite simple graph
(see [11, 15]).

######  Remark 2.6.

If $\mathcal{D}=\mathcal{D}_{1}\cup\mathcal{D}_{2}$ such that
$V(\mathcal{D}_{1})\cap V(\mathcal{D}_{2})=\emptyset$, then
$I_{\mathcal{D}}=I_{\mathcal{D}_{1}}+I_{\mathcal{D}_{2}}$. Thus we consider
graphs with only one connected component.

In a manner similar to that used in the case of unoriented graphs, we
characterize generators of the toric ideal of weighted oriented graphs in
terms of cycles and closed walks. Our characterizations depend on the
orientation of the edges in the oriented graph, and the cycle structure of its
underlying unoriented graph. As such, when we refer to a cycle in a weighted
oriented graph, we are considering a cycle in the underlying unoriented graph.
When we also want to consider the orientation of the edges on the cycle, we
use the term oriented cycle. An oriented cycle is not the same as a directed
cycle (a term commonly used in the theory of directed graphs), where the
latter requires the edges to all point in the same direction.

##  3\. Toric zero ideals

In this section, we extend Theorem 2.2, namely if a simple graph has no non-
trivial even closed walk then the toric ideal is the zero ideal. We start with
the main setting of this section which is more general than in the remainder
of the paper. Here we are associating any monomial ${\bf x}^{{\bf
a}_{i}}=x_{i_{1}}^{a_{i_{1},i}}x_{i_{2}}^{a_{i_{2},i}}$ with $i_{1}\neq i_{2}$
to the edge $\\{x_{i_{1}},x_{i_{2}}\\}$ of a graph.

######  Setting 3.1.

Let $M\subseteq k[x_{1},\dots,x_{\nu}]$ be a monomial ideal with minimal
generating set $\operatorname{gens}(M)=\\{\bf x^{\bf a_{1}},\dots,\bf x^{\bf
a_{\mu}}\\}$ such that $|\operatorname{supp}{\bf x}^{{\bf a}_{i}}|=2$ for all
${\bf x}^{{\bf a}_{i}}\in\operatorname{gens}(M)$, and whenever ${\bf x}^{{\bf
a}_{i}},{\bf x}^{{\bf a}_{j}}\in\operatorname{gens}(M)$ such that $i\neq j$
then $\operatorname{supp}{({\bf x}^{{\bf
a}_{i}})}\neq\operatorname{supp}{({\bf x}^{{\bf a}_{j}})}$. Note that
$\sqrt{M}$ is then a square-free monomial ideal of degree 2. We define the
_graph of $M$_, denoted $G(M)$, to be the graph whose edge ideal is
$\sqrt{M}$. We take $E(G(M))=\\{e_{1},\dots,e_{\mu}\\}$ so that the $I_{M}$ is
the toric ideal defined by $M$ in $k[e_{1},\dots,e_{\mu}]$.

######  Proposition 3.2.

Let $M$ and $G(M)$ be defined as in 3.1. Furthermore suppose that
$M=M_{1}+M_{2}$ where $|V(G(M_{1}))\cap V(G(M_{2}))|=1$ and $G(M_{2})$ is a
tree. Then $I_{M}=I_{M_{1}}$.

###### Proof.

Let $f=\prod_{i=1}^{\mu}e_{i}^{r_{i}}-\prod_{i=1}^{\mu}e_{i}^{s_{i}}$ be an
irreducible element in $I_{M}$. Let $E$ be the collection of all edges in the
support of $f$ that are in $G(M_{2}),$ and suppose $E\neq\emptyset$. Since
$G(M_{2})$ is a tree, the subgraph of $G(M_{2})$ with the edge set $E$ must
have a leaf, say $x_{t}.$ Let $e_{\ell}$ be the unique edge incident to
$x_{t}$ in $E$. Therefore, since
$\sum_{i=1}^{\mu}r_{i}\mathbf{a}_{i}=\sum_{i=1}^{\mu}s_{i}\mathbf{a}_{i}$,
where $\mathbf{a}_{1}\dots,\mathbf{a}_{\mu}$ are the exponent vectors of the
generators $M$, we must have $r_{\ell}=s_{\ell}\neq 0$. This contradicts the
irreducibility of $f$, and thus $E$ must in fact be empty. In other words, the
support of every irreducible generator of $I_{M}$ consists of edges of
$G(M_{1})$.
In the case that $G(M)$ is a tree we can take $M_{1}$ to be generated by a
single generator of $M$, thus giving us $I_{M}=I_{M_{1}}=\\{0\\}$. ∎

######  Corollary 3.3.

Again let $M$ and $G(M)$ be as in 3.1. Assume further that
$M=M^{\prime}+\sum_{i=1}^{k}M_{i}$ such that each $G(M_{i})$ is a tree,
$|V(G(M^{\prime}))\cap V(G(M_{i}))|=1$ for all $i=1,2,\dots,k$, and
$V(G(M_{i}))\cap V(G(M_{j}))=\emptyset$ for all $i\neq j$. Then
$I_{M}=I_{M^{\prime}}$.

######  Remark 3.4.

Since leaves do not contribute to the generators of the toric ideal by
Corollary 3.3, for the remainder of the paper, we may assume that $G(M)$ has
no leaves.

######  Proposition 3.5.

Let $M$ and $G(M)$ be as in 3.1. Suppose that $G(M)$ is an odd cycle, then
$I_{M}$ is the zero ideal.

###### Proof.

Suppose $f=\prod_{i=1}^{\mu}e_{i}^{r_{i}}-\prod_{i=1}^{\mu}e_{i}^{s_{i}}\neq
0$ is an irreducible element of $I_{M}$. For convenience, we set
$f_{+}=\prod_{i=1}^{\mu}e_{i}^{r_{i}}$ and
$f_{-}=\prod_{i=1}^{\mu}e_{i}^{s_{i}}$ and note that
$\varphi(f_{+})=\varphi(f_{-})$. Let $H$ be the subgraph of $G(M)$ induced by
the edges in the support of $f$. Note that $H$ must then either be a forest or
$H=G(M).$ In the first situation, we must have $f=0$ by Proposition 3.2. Thus,
we may assume that $H=G(M)$. Since $f$ is irreducible we have
$\operatorname{supp}(f_{+})\cap\operatorname{supp}(f_{-})=\emptyset$. Since
$H=G(M)$, we must have
$\operatorname{supp}\varphi(f)=\\{x_{1},\dots,x_{2n+1}\\}$ which in turn
implies
$\operatorname{supp}\varphi(f_{+})=\operatorname{supp}\varphi(f_{-})=\\{x_{1},\dots,x_{2n+1}\\}$.
In the cycle, each vertex $x_{i}$ is incident to exactly two edges, $e_{i-1}$
and $e_{i}$ (where we take $e_{0}=e_{2n+1}$) and so one of $e_{i-1}$ or
$e_{i}$ is in $\operatorname{supp}(f_{+})$ while the other in
$\operatorname{supp}(f_{-})$. Without loss of generality, we can take
$e_{1}\in\operatorname{supp}(f_{+})$ so that
$e_{2}\in\operatorname{supp}(f_{-})$ which in turn tells us
$e_{3}\in\operatorname{supp}(f_{+})$. Traveling around the cycle in this
manner will lead us to $e_{2n+1}\in\operatorname{supp}(f_{+})$. Since both
edges incident to $x_{1}$ are then in $\operatorname{supp}(f_{+})$,
$x_{1}\not\in\operatorname{supp}\varphi(f_{-})$ thus leading to a
contradiction. ∎

We are now ready to state the main theorem of this section.

######  Theorem 3.6.

Let $M$ and $G(M)$ be defined as in 3.1 such that $G(M)$ has only trivial even
closed walks. Then the toric ideal $I_{M}$ is the zero ideal.

###### Proof.

Saying that $G(M)$ has only trivial even closed walks means that each
connected component of $G(M)$ can contain at most one cycle, which must be of
odd length. By Remark 2.6 and Corollary 3.3 we may then assume that $G(M)$ is
an oriented odd cycle. Now the theorem follows from Proposition 3.5. ∎

##  4\. Unicyclic graphs

In the remainder of this paper, we turn our attention to finding a generating
set for the toric ideal of a weighted oriented graph as defined in Section 2.
We start by analyzing those graphs with a single cycle. This is not a
straightforward generalization of the unoriented case as the orientations of
the edges of the cycle and the weights affect the generators of the ideal.

Let $\mathcal{C}_{n}$ be a weighted oriented graph whose underlying unoriented
graph is a cycle of length $n$. Then the incidence matrix $A(\mathcal{C}_{n})$
is an $n\times n$ matrix of the following form:

$A(\mathcal{C}_{n})=\left[\begin{array}[]{cccccccc}a_{1,1}&0&&\dots&&&0&a_{1,n}\\\
a_{2,1}&a_{2,2}&&\dots&&&0&0\\\ 0&a_{3,2}&\ddots\\\ &&\ddots&a_{i,i}\\\
\vdots&\vdots&&a_{i+1,i}&&&\vdots&\vdots\\\ &&&&&\ddots\\\
0&0&\dots&&&\ddots&a_{n-1,n-1}&0\\\
0&0&\dots&&&0&a_{n,n-1}&a_{n,n}\end{array}\right]$

where $a_{i,i-1},a_{i,i}\in\\{1,w_{i}\\}$ under the convention that
$a_{1,0}=a_{1,n}$. We observe that if $\det(A(\mathcal{C}_{n}))\neq 0$, then
the matrix has full rank. This implies the toric ideal, $I_{\mathcal{C}_{n}}$
is zero. On the other hand, if
$\det(A(\mathcal{C}_{n}))=\prod_{i=1}^{n}a_{i,i}+(-1)^{n+1}a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}=0$,
then $\text{Null}(A(\mathcal{C}_{n}))$ is non-zero and there is at least one
generator in $I_{\mathcal{C}_{n}}.$ Due to the cycle structure, any generator
in $I_{\mathcal{C}_{n}}$ must involve all edges of the cycle which we now
prove.

######  Lemma 4.1.

If $f$ is a non-zero element of $I_{\mathcal{C}_{n}}$, then
$\operatorname{supp}(f)=E(\mathcal{C}_{n})$.

###### Proof.

Let $0\neq f\in I_{\mathcal{C}_{n}}$ and $\textbf{u}=(u_{1},\ldots,u_{n})$ its
corresponding non-trivial element in $\text{Null}(A(\mathcal{C}_{n}))$ so that
$A(\mathcal{C}_{n}){\bf u}=\sum_{i=1}^{n}u_{i}\textbf{a}_{i}=\textbf{0}$ where
$\textbf{a}_{i}$ is the $i^{\text{th}}$ column of $A(\mathcal{C}_{n})$. We
claim that $u_{i}\neq 0$ for all $i\in[n]$ from which the lemma immediately
follows. For the sake of contradiction suppose there exists $e_{j}\in
E(\mathcal{C}_{n})$ such that $u_{j}=0$, and let $x_{\ell}$ be a vertex
incident to edge $e_{j}$ in $\mathcal{C}_{n}$. Up to a relabeling of the
graph, we may assume that $\ell\neq n$. Then let $e_{j^{\prime}}$ be the only
other edge incident to $x_{\ell}$ in $\mathcal{C}_{n}$. Since the
$\ell^{\text{th}}$ component of the vector $A(\mathcal{C}_{n}){\bf u}$
satisfies $(A(\mathcal{C}_{n}){\bf
u})_{\ell}=\sum_{i=1}^{n}u_{i}a_{\ell,i}=0$, and by assumption $u_{i}=0$ for
all $i\neq j^{\prime}$, we must have $u_{j^{\prime}}=0$ as well. An iterative
application of this argument results in $u_{i}=0$ for all $e_{i}\in
E(\mathcal{C}_{n}),$ which contradicts the fact that
$\textbf{u}=(u_{1},\ldots,u_{n})$ is a non-trivial element in
$\text{Null}(A(\mathcal{C}_{n}))$. ∎

By Lemma 4.1, if $\det(A(\mathcal{C}_{n}))=0$, there exists a minimal
generator $f$ of $I_{\mathcal{C}_{n}}$ such that $\operatorname{supp}f$
involves all edges of the cycle. In order to have
$\det(A(\mathcal{C}_{n}))=\prod_{i=1}^{n}a_{i,i}+(-1)^{n+1}a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}=0$,
$n$ must be an even number and
$\prod_{i=1}^{n}a_{i,i}=a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}$. This motivates the
following definition.

######  Definition 4.2.

If $\mathcal{C}_{n}$ is a weighted oriented cycle on $n$ vertices and
$A(\mathcal{C}_{n})$ is its incidence matrix, we say $\mathcal{C}_{n}$ is
_balanced_ if $n$ is even and
$\prod_{i=1}^{n}a_{i,i}=a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}$.

######  Theorem 4.3.

Let $\mathcal{C}_{n}$ be a weighted oriented cycle on $n$ vertices. Then the
toric ideal $I_{\mathcal{C}_{n}}$ is non-zero if and only if $\mathcal{C}_{n}$
is balanced.

###### Proof.

As noted above, the toric ideal $I_{\mathcal{C}_{n}}\neq 0$ if and only if the
incidence matrix $A(\mathcal{C}_{n})$ does not have full rank, or equivalently
if and only if $\det(A(\mathcal{C}_{n}))=0$. Therefore
$I_{\mathcal{C}_{n}}\neq 0$ if and only if

$\prod_{i=1}^{n}a_{i,i}+(-1)^{n+1}a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}=0.$

Or equivalently

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

Since all $a_{i,i},a_{i+1,i},a_{1,n}\geq 1$, this last equation can hold if
and only if $n$ is even and
$\prod_{i=1}^{n}a_{i,i}=a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}$. ∎

The next result follows immediately from either Theorem 4.3 or Proposition
3.5.

######  Corollary 4.4.

If $\mathcal{C}_{n}$ is a weighted oriented cycle on an odd number of
vertices, then $I_{\mathcal{C}_{n}}=0$.

The following algorithm produces the single generator of the toric ideal of a
balanced cycle.

######  Algorithm 4.5.

Input: $\mathcal{C}_{n}$ a balanced oriented weighted cycle with edges
$e_{1},\dots,e_{n}$ with weight vector $\textbf{w}=(w_{1}\dots w_{n})$. The
algorithm outputs the generator of $I_{\mathcal{C}_{n}}$.

  * Step 0:

Set $i=1$ and $r_{1}=1$.

  * Step 1:

For $2\leq i\leq n$,

if $e_{i-1}=(x_{i},x_{i-1})$ and $e_{i}=(x_{i+1},x_{i})$, then
$r_{i}=\frac{r_{i-1}}{w_{i}}$

else if $e_{i-1}=(x_{i-1},x_{i})$ and $e_{i}=(x_{i},x_{i+1})$, then
$r_{i}=w_{i}r_{i-1}$

else $r_{i}=r_{i-1}$.

  * Step 2:

If no $r_{i}$ is fractional, then set $b_{i}=r_{i}$ for $1\leq i\leq n$

else, $b_{i}=d*r_{i}$ where $d$ is the least common multiple of the
denominators of $r_{1},...,r_{n}$ in reduced form.

Output: $e_{1}^{b_{1}}e_{3}^{b_{3}}\dots
e_{n-1}^{b_{n-1}}-e_{2}^{b_{2}}e_{4}^{b_{4}}\dots e_{n}^{b_{n}}$.

######  Remark 4.6.

Note that the output binomial of Algorithm 4.5 is indeed an element of the
toric ideal. The construction forces the exponent vector of each monomial of
$\varphi(e_{1}^{b_{1}}e_{3}^{b_{3}}\dots
e_{n-1}^{b_{n-1}}-e_{2}^{b_{2}}e_{4}^{b_{4}}\dots e_{n}^{b_{n}})$ to have the
same value at each variable. The positions of the non-zero entries of the
incidence matrix $A(\mathcal{C}_{n})$ guarantee that the rank of
$A(\mathcal{C}_{n})$ is at least $n-1$, whereas the definition of a balanced
cycle requires $\det(A(\mathcal{C}_{n}))=0$. Hence the nullspace of
$A(\mathcal{C}_{n})$ has dimension 1. Since we obtained the vector
$(b_{1},\dots,b_{n})$ by multiplying by the least common multiple of the
denominators, every integer-valued vector in the nullspace is a scalar
multiple of $(b_{1},\dots,b_{n})$.

######  Example 4.7.

Let $\mathcal{D}_{1}$ be the weighted 8-cycle with edges
$(x_{1},x_{2}),(x_{2},x_{3}),(x_{3},x_{4}),(x_{5},x_{4}),(x_{6},x_{5}),\\\
(x_{7},x_{6}),(x_{8},x_{7}),(x_{1},x_{8})$ and weight vector, ${\bf
w}=(1,4,4,3,2,2,2,2)$. The toric ideal of $\mathcal{D}_{1}$ is

$I_{\mathcal{D}_{1}}=(e_{1}e_{3}^{16}e_{5}^{8}e_{7}^{2}-e_{2}^{4}e_{4}^{16}e_{6}^{4}e_{8}).$

Let $\mathcal{D}_{2}$ be the weighted 8-cycle with edges
$(x_{1},x_{2}),(x_{2},x_{3}),(x_{3},x_{4}),(x_{5},x_{4}),(x_{6},x_{5}),\\\
(x_{7},x_{6}),(x_{8},x_{7}),(x_{1},x_{8})$ and weight vector, ${\bf
w}=(1,8,2,3,2,2,2,2)$. The toric ideal of $\mathcal{D}_{2}$ is

$I_{\mathcal{D}_{2}}=(e_{1}e_{3}^{16}e_{5}^{8}e_{7}^{2}-e_{2}^{8}e_{4}^{16}e_{6}^{4}e_{8}).$

Let $\mathcal{D}_{3}$ be the weighted 8-cycle with edges
$(x_{1},x_{2}),(x_{2},x_{3}),(x_{4},x_{3}),(x_{5},x_{4}),(x_{6},x_{5}),\\\
(x_{7},x_{6}),(x_{8},x_{7}),(x_{1},x_{8})$ and weight vector, ${\bf
w}=(1,48,4,3,2,2,2,2)$.

The toric ideal of $\mathcal{D}_{3}$ is

$I_{\mathcal{D}_{3}}=(e_{1}e_{3}^{48}e_{5}^{8}e_{7}^{2}-e_{2}^{48}e_{4}^{16}e_{6}^{4}e_{8}).$

The weighted oriented graphs $\mathcal{D}_{1}$, $\mathcal{D}_{2}$, and
$\mathcal{D}_{3}$ are all balanced. The graphs $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ differ only by the weights on $x_{2}$ and $x_{3}$. We see
this difference manifests in their respective generators, specifically the
exponent on $e_{2}$. The differences between $\mathcal{D}_{1}$ and
$\mathcal{D}_{3}$ are the orientation of the edge on $\\{x_{3},x_{4}\\}$ and
the weight of $x_{2}$. Here we see this difference in the generators via the
exponents on $e_{3}$ and $e_{2}$.

From Example 4.7, we see that in general, the condition for a cycle to be
balanced is a complex interaction between the weights on the vertices and the
orientations of the edges. We consider the extreme cases (when the edges are
oriented in the same direction and when the weights on the vertices are all
the same) in the next two corollaries.

We first consider the case where all of the edges of the cycle are oriented in
the same direction. Formally, we say that a cycle
$\\{x_{1},x_{2}\\},\\{x_{2},x_{3}\\},\dots\\{x_{n},x_{1}\\}$ is _naturally
oriented_ if the orientations on the edges are either all $(x_{i},x_{i+1})$ or
all $(x_{i+1},x_{i})$, where we take $x_{n+1}=x_{1}$.

######  Corollary 4.8.

Let $\mathcal{D}$ be a weighted oriented graph with a unique cycle that is
naturally oriented on $n$ vertices with $w_{i}\geq 2$ for some $i\in[n]$. Then
$I_{\mathcal{D}}$ is the zero ideal.

###### Proof.

First, suppose that $\mathcal{D}$ consists of a single cycle. If $\mathcal{D}$
is an odd cycle, then by Corollary 4.4 $I_{\mathcal{D}}$ must be the zero
ideal regardless of the orientation of the edges. Let $\mathcal{D}$ be a
naturally oriented even cycle. Then $\det
A(\mathcal{D})=\prod_{i=1}^{n}a_{i,i}-a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}=\pm(1-\prod_{i=1}^{n}w_{i})\neq
0,$ as $w_{i}>1$ for some $i\in[n]$. Thus $\mathcal{D}$ is not balanced, and
by Theorem 4.3 $I_{\mathcal{D}}$ must be the zero ideal. The full result then
follows from Corollary 3.3. ∎

We next consider oriented cycles for which the weights are all the same. For
this, we will want to consider graphs in which half the edges are oriented in
a clockwise manner and half counter-clockwise. To make this precise we
introduce the following definition.

######  Definition 4.9.

Let $\mathcal{C}_{n}$ be a weighted oriented graph with its underlying graph
the cycle on $n$ vertices. Assume that $n$ is even. Let
$E(\mathcal{C}_{n})=\\{e_{1},\dots,e_{n}\\}$ such that for all $i=1,\dots,n$,
$e_{i}=\\{x_{i},x_{i+1}\\}$ where we take $x_{n+1}=x_{1}$. We then define the
following subsets of $E(\mathcal{C}_{n})$.

$\displaystyle\mathcal{C}_{n}^{inc}$
$\displaystyle=\\{e_{i}=(x_{i},x_{i+1})~{}:~{}i=1,\dots,n\\}$
$\displaystyle\mathcal{C}_{n}^{dec}$
$\displaystyle=\\{e_{i}=(x_{i+1},x_{i})~{}:~{}i=1,\dots,n\\}=E(\mathcal{C}_{n})\setminus\mathcal{C}_{n}^{inc}$

The oriented even cycle $\mathcal{C}_{n}$ is called _uniformly balanced_ if
and only if $|\mathcal{C}_{n}^{inc}|=|\mathcal{C}_{n}^{dec}|$ and $w_{i}=w$
for all $1\leq i\leq n$ such that the vertex $x_{i}\in V(\mathcal{C}_{n})$ is
neither a sink nor a source.

######  Corollary 4.10.

Let $\mathcal{C}_{n}$ be a weighted oriented cycle on $n$ vertices such that
$w_{i}=w\geq 2$ for all non-source and non-sink $x_{i}\in V(\mathcal{C}_{n})$.
The toric ideal $I_{\mathcal{C}_{n}}$ is non-zero if and only if
$\mathcal{C}_{n}$ is uniformly balanced.

###### Proof.

Let $\mathcal{C}_{n}$ be a weighted oriented cycle on $n$ vertices in which
$w_{i}=w\geq 2$ for all non-source and non-sink vertices $x_{i}$. Let
$A(\mathcal{C}_{n})$ be the incidence matrix of $\mathcal{C}_{n}$. By Theorem
4.3, $I_{\mathcal{C}_{n}}\neq 0$ if and only if
$\prod_{i=1}^{n}a_{i,i}=a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}$ and $n$ is even. By
assumption, $\mathcal{C}_{n}$ is uniformly balanced and thus $n$ is even, so
we just need to verify the equality. Notice that if $x_{p}$ is a sink vertex
with weight $w_{p}$, the $p$-th row of $A(\mathcal{C}_{n})$ has
$a_{p,p-1}=w_{p}$ and $a_{p,p}=w_{p}$. Hence the sink vertex does not impact
the equality $\prod_{i=1}^{n}a_{i,i}=a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}$.
Similarly, if $x_{p}$ is a source then $a_{p,p-1}=1$ and $a_{p,p}=1$. By
assumption all non-source and non-sink vertices have the same weight. In the
$i$-th column of the $A(\mathcal{C}_{n})$ we have two possibilities: either
$a_{i,i}=w$ and $a_{i+1,i}=1$ or vice versa, again taking $a_{n+1,n}=a_{1,n}$.
In the first case, the edge $e_{i}=(x_{i+1},x_{i})\in\mathcal{C}_{n}^{dec}$
and in the second $e_{i}=(x_{i},x_{i+1})\in\mathcal{C}_{n}^{inc}$. Thus
$\prod_{i=1}^{n}a_{i,i}=a_{1,n}\prod_{i=1}^{n-1}a_{i+1,i}$ if and only if half
of the edges are in $\mathcal{C}_{n}^{inc}$ and half are in
$\mathcal{C}_{n}^{dec}$. Thus $I_{\mathcal{C}_{n}}\neq 0$ if and only if
$\mathcal{C}_{n}$ is uniformly balanced. ∎

######  Example 4.11.

Let $\mathcal{D}_{1}$ be the weighted 8-cycle with edges
$(x_{1},x_{2}),(x_{2},x_{3}),(x_{4},x_{3}),(x_{5},x_{4}),(x_{6},x_{5}),\\\
(x_{6},x_{7}),(x_{7},x_{8}),(x_{1},x_{8})$ and with all vertices with weight
3. The toric ideal of $\mathcal{D}_{1}$ is then

$I_{\mathcal{D}_{1}}=(e_{1}^{3}e_{3}^{9}e_{5}e_{7}^{3}-e_{2}^{9}e_{4}^{3}e_{6}e_{8}^{3}).$

Let $\mathcal{D}_{2}$ be the weighted 8-cycle with edges
$(x_{1},x_{2}),(x_{2},x_{3}),(x_{4},x_{3}),(x_{5},x_{4}),(x_{5},x_{6}),\\\
(x_{6},x_{7}),(x_{7},x_{8}),(x_{1},x_{8})$ also with all vertices having
weight 3. The toric ideal of $\mathcal{D}_{2}$ is then the zero ideal. The
only difference between $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ is the
orientation of the edge on vertices $\\{x_{5},x_{6}\\}$ so that
$|\mathcal{D}_{2}^{inc}|\neq|\mathcal{D}_{2}^{dec}|$.

##  5\. Two Connected Cycles

We now consider generating sets of the toric ideals of connected oriented
graphs comprised of distinct oriented cycles $\mathcal{C}_{m}$ and
$\mathcal{C}_{n}$ with varying degrees of overlap. More specifically, we
determine for which of these graphs the toric ideal can be generated by a
single element. We then give a combinatorial description of the support of
that generator.

Before we state the main theorem of this section, we introduce one additional
notation. Let ${\mathcal{P}}_{s}$ denote an oriented path of length $s$, i.e.
its underlying unoriented graph is a path. Note that the incidence matrix of
${\mathcal{P}}_{s}$ is of size $(s+1)\times s$ and has the following form:

$A(\mathcal{P}_{s})=\left[\begin{array}[]{ccccccc}a_{1,1}&0&&\dots&&&0\\\
a_{2,1}&a_{2,2}&&\dots&&&0\\\ 0&a_{3,2}&\ddots\\\ &&\ddots&a_{i,i}\\\
\vdots&\vdots&&a_{i+1,i}&&&\vdots\\\ &&&&&\ddots\\\
0&0&\dots&&&\ddots&a_{s,s}\\\ 0&0&\dots&&&0&a_{s+1,s}\end{array}\right].$

######  Theorem 5.1.

Let $\mathcal{D}$ be a weighted oriented graph comprised of two oriented
cycles $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ satisfying one of the
following:

  1. (A)

$\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ share a single vertex,

  2. (B)

the set $E(\mathcal{C}_{m})\cap E(\mathcal{C}_{n})$ induces an oriented path
with at least one edge, or

  3. (C)

$\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ are connected by an oriented path
${\mathcal{P}}_{s}$ of length $s\geq 1$.

Then the toric ideal $I_{\mathcal{D}}$ is generated by a single element if and
only if at most one of its cycles is balanced.

Figure 1 illustrates the three possible forms of the underlying unoriented
cycle structure of $\mathcal{D}$ along with a labeling of vertices and edges
that will be used throughout this section.

(a) Two cycles sharing a vertex

(b) Two cycles sharing edges

(c) Two cycles connected by a path

Figure 1. Cycle forms from Theorem 5.1

(a) Two cycles sharing a vertex

(b) Two cycles sharing edges

(c) Two cycles connected by a path

Figure 2. Incidence matrix forms from Theorem 5.1

######  Remark 5.2.

Note that in Figure 1(B) there are three distinct cycles to consider in
$\mathcal{D}$: the two “inner cycles” $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$,
and the “outer” cycle obtained by deleting the edges
$\\{e_{m-k+1},\dots,e_{m}\\}$. For $I_{\mathcal{D}}$ to be generated by a
single element, at most one of these three cycles can be balanced.

###### Proof.

If both $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ are balanced then Corollary
4.10 tells us that their corresponding toric ideals are nonzero. Using a
straightforward linear algebra argument, one can extend a basis element of
$\operatorname{Null}A(\mathcal{C}_{m})$ and a basis element of
$\operatorname{Null}A(\mathcal{C}_{n})$ to linearly independent elements of
$\operatorname{Null}A(\mathcal{D})$. Thus $I_{\mathcal{D}}$ would have at
least two generators. The same argument would apply in the case of overlapping
edges for which two of the three cycles in $\mathcal{D}$ are balanced, see
Remark 5.2.

We now assume that $\mathcal{D}$ has at most one balanced cycle, and
furthermore suppose that $\mathcal{C}_{m}$ is not balanced. Throughout each of
the three cases, we refer to the $(i,j)^{\text{th}}$ entry of $A(\mathcal{D})$
as $a_{i,j}$. We also define $\widetilde{A(\mathcal{D})}$ to be the square
matrix obtained from the incidence matrix $A(\mathcal{D})$ by deleting the
last column. Informed by the labeling of vertices and edges as in Figure 1,
the matrices in Figure 2 illustrate the possible forms of the incidence matrix
of $\mathcal{D}$. Note that in all three situations, $A(\mathcal{D})$ has one
more column than row so that $\widetilde{A(\mathcal{D})}$ is square. Thus if
$\det\widetilde{A(\mathcal{D})}\neq 0$ then $A(\mathcal{D})$ is full rank and
$\dim_{\mathbb{Q}}\operatorname{Null}A(\mathcal{D})=1$. This in turn shows
$I_{\mathcal{D}}$ is principal.

We first consider the case that $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ are
joined by a single vertex. Define $B$ to be the matrix obtained from
$A(\mathcal{C}_{n})$ by deleting both the first row and the last column. Then
$\det\widetilde{A(\mathcal{D})}=\det A(\mathcal{C}_{m})\det B$. Since
$\mathcal{C}_{m}$ is not balanced we must have that $\det
A(\mathcal{C}_{m})\neq 0$. One can also calculate

$\det B=\prod_{i=m+1}^{m+n-1}a_{i,i}\neq 0$

where $a_{i,i}\in\\{1,w_{i}\\}$ for all $i$. Thus $A(\mathcal{D})$ is of full
rank and $I_{\mathcal{D}}$ has a single generator.

We now consider the case that $E(\mathcal{C}_{m})\cap E(\mathcal{C}_{n})$
induces a path of length $k\geq 1$. Note that in the block formation of
$A(\mathcal{D})$ as illustrated in Figure 2(B) the entries below the overlap,
i.e. the $(i,j)$-entries with $i\geq m+1$ and $m-k\leq j\leq m$, are all zero.
Therefore, $\det\widetilde{A(\mathcal{D})}=\det A(\mathcal{C}_{m})\det B_{k}$
where $B_{k}$ is obtained from $A(\mathcal{C}_{n})$ by deleting its first $k$
columns, its first $k+1$ rows, and the last column. It can then be observed
that

$\det B_{k}=\prod_{i=m+1}^{m+n-k-1}a_{i,i}\neq 0.$

Again we see that $A(\mathcal{D})$ is indeed of full rank.

Lastly, assume $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ are connected by an
oriented path ${\mathcal{P}}_{s}$ with $s\geq 2$. Again define the matrix $B$
to be the matrix obtained from $A(\mathcal{C}_{n})$ by deleting the first row
and last column. We also take $E$ to be the matrix obtained from
$A({\mathcal{P}}_{s})$ by deleting its first row. Note that the
$(i,j)^{\text{th}}$ entry of $A({\mathcal{P}}_{s})=0$ if and only if $j>i+1$
or $j<i$. Thus $\det\widetilde{A(\mathcal{D})}=\det A(\mathcal{C}_{m})\det
E\det B\neq 0$ since

$\det E=\prod_{i=m+1}^{m+s}a_{i,i}\neq 0.$

Yet again we see that $A(\mathcal{D})$ is of full rank. ∎

In the following corollary, we give a combinatorial description of the
generator guaranteed by Theorem 5.1 in terms of the edges appearing in the
support of the generator. Again we will see that there are three cycles to
consider in the case that $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ share edges.

######  Corollary 5.3.

Let $\mathcal{D}$ be an oriented graph comprised of two cycles as in Theorem
5.1 such that $I_{\mathcal{D}}$ is generated by a single irreducible binomial
$f$. Then $\operatorname{supp}f=E(\mathcal{C})$ if one of the cycles
$\mathcal{C}$ is balanced. Otherwise, $\operatorname{supp}f=E(\mathcal{D})$.

###### Proof.

Let $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ be distinct cycles in
$\mathcal{D}$ where $I_{\mathcal{D}}$ is generated by a single nonzero element
$f$. By Theorem 5.1 we know at least one of these cycles is unbalanced, assume
it is $\mathcal{C}_{n}$. If $\mathcal{C}_{m}$ is balanced, then we can
trivially extend ${\bf u}\in\operatorname{Null}A(\mathcal{C}_{m})$ to an
element of $\operatorname{Null}A(\mathcal{D})$ so that by Lemma 4.1 we can
assume the generator $f$ of $I_{\mathcal{D}}$ satisfies
$\operatorname{supp}f=E(\mathcal{C}_{m})$. Now assume that $\mathcal{C}_{m}$
is also unbalanced.

We first argue that $E(\mathcal{C}_{m})\setminus
E(\mathcal{C}_{n})\subseteq\operatorname{supp}f.$ Let ${\bf
u}\in\operatorname{Null}A(\mathcal{D})$ so that
$\sum_{i=1}^{|E(\mathcal{D})|}u_{i}{\bf a}_{i}=\bf 0$, and for the sake of
contradiction suppose $u_{j}=0$ for some $e_{j}\in E(\mathcal{C}_{m})\setminus
E(\mathcal{C}_{n})$. Since $\mathcal{D}$ has no multiple edges, we can find
$x_{\ell}\in V(\mathcal{C}_{m})\setminus V(\mathcal{C}_{n})$ incident to
$e_{j}$. There is then only one other edge incident to $x_{\ell}$ which we can
assume to be $e_{i+1}\in E(\mathcal{C}_{m})$. It follows then that
$u_{i+1}=0$. Iterating this argument yields $u_{i}=0$ for all $e_{i}\in
E(\mathcal{C}_{m})\setminus E(\mathcal{C}_{n})$ which would imply that
$\operatorname{supp}f\subseteq E(\mathcal{D})\setminus E(\mathcal{C}_{m})$.
Note that the only cycle in $\mathcal{D}\setminus\mathcal{C}_{m}$ is
$\mathcal{C}_{n}$ so that $I_{\mathcal{D}}=I_{\mathcal{C}_{n}}$ by Proposition
3.2. However, since $\mathcal{C}_{n}$ is unbalanced Theorem 4.3 yields
$I_{\mathcal{D}}=I_{\mathcal{C}_{n}}$ is the zero ideal, a contradiction.
Since $\mathcal{C}_{m}$ is also unbalanced, a symmetric argument tells us we
also have $E(\mathcal{C}_{n})\setminus
E(\mathcal{C}_{m})\subseteq\operatorname{supp}f$.

In the case that $\mathcal{C}_{m}\cap\mathcal{C}_{n}$ is a single vertex we
have

$E(\mathcal{C}_{m})\bigtriangleup E(\mathcal{C}_{n})=E(\mathcal{C}_{m})\cup
E(\mathcal{C}_{n})=E(\mathcal{D})$

where $E(\mathcal{C}_{m})\bigtriangleup E(\mathcal{C}_{n})$ denotes the
symmetric difference of $E(\mathcal{C}_{m})$ and $E(\mathcal{C}_{n})$. Thus we
must have $\operatorname{supp}f=E(\mathcal{D})$.

When $\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ are connected by a path of length
at least one, we again see that

$E(\mathcal{C}_{m})\bigtriangleup E(\mathcal{C}_{m})=E(\mathcal{C}_{m})\cup
E(\mathcal{C}_{n})\subseteq\operatorname{supp}f.$

Let $\mathcal{D}$ be labeled as in Figure 1(C). Note that since
$\mathcal{C}_{m}$ and $\mathcal{C}_{n}$ are unbalanced, ${\bf
u}\in\operatorname{Null}A(\mathcal{D})$ would have to satisfy $u_{i}\neq 0$
for some $m+1\leq i\leq m+s$, otherwise $I_{\mathcal{C}_{m}}\neq(0)\neq
I_{\mathcal{C}_{n}}$. In the case that $s=1$, then we would immediately have
$\operatorname{supp}f=E(\mathcal{D})$. Suppose that $s\geq 2$ and $u_{j}\neq
0$ for some $e_{j}\in{\mathcal{P}}_{s}$. Up to a relabeling of $\mathcal{D}$
we can assume that $x_{j}\in
V({\mathcal{P}}_{s})\setminus[V(\mathcal{C}_{m})\cup V(\mathcal{C}_{n})]$ for
which the only other edge incident to it is $e_{j+1}\in E({\mathcal{P}}_{s})$.
Since $\sum u_{i}{\bf a}_{i}=\bf{0}$ and $u_{j}\neq 0$ we must then have
$u_{j+1}\neq 0$. Iterating this argument results in
$E({\mathcal{P}}_{s})\subseteq\operatorname{supp}f$ so that again
$\operatorname{supp}f=E(\mathcal{D})$.

Lastly, we consider the case that $E(\mathcal{C}_{m})\cap E(\mathcal{C}_{n})$
induces a path of length $k$, call it ${\mathcal{P}}_{k}$. If the outer cycle
induced by $E(\mathcal{C}_{m})\bigtriangleup E(\mathcal{C}_{n})$ is balanced,
we can then assume that $f$ is supported by this cycle. If the outer cycle is
unbalanced, then we must have $u_{i}\neq 0$ for some
$e_{i}\in{\mathcal{P}}_{k}$. By a similar argument as above, we can then
conclude that $u_{i}\neq 0$ for every $e_{i}\in{\mathcal{P}}_{k}$ which again
results in $f$ being supported by all of the edges in $\mathcal{D}$.

∎

######  Remark 5.4.

When constructing the toric ideal of an unoriented graph, one can construct a
binomial generator $f$ from an even closed walk by assigning alternating edges
along the walk to each monomial in $f$. One can similarly think of
constructing the generator of the toric ideal $I_{\mathcal{D}}$ where
$\mathcal{D}$ is of one of the forms in Theorem 5.1. However, the parity of
the lengths of the cycles will affect in which of the monomials each edge
appears. Furthermore, due to the weights on the vertices, it is difficult to
calculate the exponents of each variable.

The following example illustrates different cases of Theorem 5.1 and Corollary
5.3. We further discuss how the generating set of the toric ideal differs from
that of the underlying graph.

(a) $\mathcal{D}_{1}$: Two unbalanced cycles

(b) $\mathcal{D}_{2}$: One balanced cycle

(c) $\mathcal{D}_{3}$: One balanced cycle

Figure 3. Graphs with two cycles

######  Example 5.5.

We see in Figure 3(A) $\mathcal{D}_{1}$ is comprised of an oriented
$\mathcal{C}_{6}$ and an oriented $\mathcal{C}_{5}$ connected by a vertex.
With the weight vector ${\bf w}=(2,2,3,4,6,1,4,2,3,2)$, neither of these
cycles are balanced. In this case

$I_{\mathcal{D}_{1}}=(e_{2}^{52}e_{4}^{156}e_{6}^{13}e_{7}^{12}e_{9}^{6}e_{11}-e_{1}^{26}e_{3}^{156}e_{5}^{26}e_{8}^{12}e_{10}^{2}).$

Here the toric ideal of the underlying graph would also be generated by a
single element, but it would come from the 6-cycle, not the whole graph.

In Figure 3(B), $\mathcal{D}_{2}$ is an oriented $\mathcal{C}_{4}$ and an
oriented $\mathcal{C}_{4}$ sharing one edge with weight vector ${\bf
w}=(2,2,3,1,2,2)$. Both 4-cycles are unbalanced but the outer 6-cycle is
balanced. Thus the 6-cycle admits the single generator,

$I_{\mathcal{D}_{2}}=(e_{1}e_{3}^{4}e_{6}^{2}-e_{2}^{2}e_{5}e_{7}^{4}).$

In the case of the underlying graph, there would be two generators, each
coming from one of the 4-cycles in the graph. Of course, the outer 6-cycle
would still admit a binomial in the toric ideal, but it would not be needed as
a minimal generator.

In Figure 3(C), $\mathcal{D}_{3}$ is comprised of two oriented 6-cycles
connected by a path of length 2. With the weight vector ${\bf
w}=(2,2,3,4,12,1,5,3,2,3,5,4,2)$ only the left cycle is balanced, and thus
gives us the generator of $I_{\mathcal{D}_{3}}$.

$I_{\mathcal{D}_{3}}=(e_{2}^{4}e_{4}^{12}e_{6}-e_{1}^{2}e_{3}^{12}e_{5})$

In contrast, the toric ideal of the underlying graph of $\mathcal{D}_{3}$
would have two minimal generators that could be taken to be from each of the
two 6-cycles.

##  6\. Toric ideals generated by a single element

The following theorem is stated as generally as possible, allowing for trees
to be attached to the graph that can be removed without affecting the toric
ideal by Proposition 3.2. Note that the graph structures described in Theorem
6.1 are the same as those in Theorem 5.1.

######  Theorem 6.1.

Let $\mathcal{D}$ be a weighted oriented graph for which the toric ideal
$I_{D}$ is generated by a single element. Then $\mathcal{D}$ is one of the
following:

  1. (1)

a balanced unicyclic graph,

  2. (2)

a graph with exactly two oriented cycles connected by either a vertex or a
path such that at most one of those cycles is balanced, or

  3. (3)

a graph with exactly three cycles such that the intersection of any two of the
cycles is a connected path and at most one of those cycles is balanced.

###### Proof.

Let $\mathcal{D}$ be a weighted oriented graph whose toric ideal
$I_{\mathcal{D}}$ is generated by a single element. Let $A(\mathcal{D})$ be
the incidence matrix of $\mathcal{D}$. In order for the null space of
$A(\mathcal{D})$ to have dimension 1, we must have
$|E(\mathcal{D})|-|V(\mathcal{D})|\leq 1$.

Let $T$ be a spanning tree of $\mathcal{D}$. It is well-known that
$|E(T)|-|V(T)|=-1$. We know that $T\neq\mathcal{D}$ since the toric ideal of a
weighted oriented tree is 0. Therefore the difference between $\mathcal{D}$
and $T$ is either one or two edges. If $\mathcal{D}$ has one more edge than
$T$, then $\mathcal{D}$ is a unicyclic graph.

If $\mathcal{D}$ has two more edges than $T$, then as above adding the first
edge to $T$ gives a unicyclic graph. Let $\\{v,w\\}$ be the second edge. If
$v$ and $w$ are both on the cycle, then $\mathcal{D}$ is a graph with two
cycles that share an edge. If $v$ is on the cycle and $w$ is not then
$\mathcal{D}$ is two cycles sharing some number of edges or two cycles sharing
a vertex. If $v$ and $w$ are both not on the cycle then $\mathcal{D}$ is
either two cycles sharing some number of edges or two cycles sharing a vertex
or two cycles connected by a bridge. ∎

We conclude with an example that illustrates the complex behavior of the
generators of toric ideals of more complicated weighted oriented graphs.

######  Example 6.2.

Let $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ be the weighted oriented graphs
with 3 cycles connected by a vertex as shown in Figure 4.

Figure 4. Three cycles joined at a vertex

If $\mathcal{D}_{1}$ has weight vector ${\bf w}=(2,2,2,2,1,2,1,2)$, then the
toric ideal of $\mathcal{D}_{1}$ is

$I_{\mathcal{D}_{1}}=\left(\begin{array}[]{lll}e_{4}e_{6}e_{7}e_{9}-e_{5}e_{8}e_{10}^{2},&e_{1}e_{3}^{4}e_{5}^{3}-e_{2}^{2}e_{4}^{3}e_{6}^{3},&e_{1}e_{3}^{4}e_{5}^{2}e_{7}e_{9}-e_{2}^{2}e_{4}^{2}e_{6}^{2}e_{8}e_{10}^{2},\\\
e_{1}e_{3}^{4}e_{5}e_{7}^{2}e_{9}^{2}-e_{2}^{2}e_{4}e_{6}e_{8}^{2}e_{10}^{4},&e_{1}e_{3}^{4}e_{7}^{3}e_{9}^{3}-e_{2}^{2}e_{8}^{3}e_{10}^{6}\end{array}\right).$

When $\mathcal{D}_{2}$ has weight vector ${\bf w}=(2,2,2,2,1,2,1,2)$, the
toric ideal of $\mathcal{D}_{2}$ is

$I_{\mathcal{D}_{2}}=(e_{7}^{2}e_{9}-e_{8}e_{10}^{2},e_{1}e_{3}^{4}e_{5}^{3}-e_{2}^{2}e_{4}^{3}e_{6}^{3}).$

The only difference between $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ is the
orientation of the edge $\\{x_{1},x_{6}\\}$. In $\mathcal{D}_{2}$ the 4-cycle
is balanced but the 4-cycles is not in $\mathcal{D}_{1}$. We observe that any
two cycles in $\mathcal{D}_{1}$ give a single generator, and there are two
generators having all edges as their supports in $I_{\mathcal{D}_{1}}$. By
contrast, $\mathcal{D}_{2}$ and the underlying graph of $\mathcal{D}_{1}$ only
have two generators, one coming from the two 3-cycles, and the other from the
4-cycle.

This example shows that even though the weights of the weighted oriented
graphs are the same, the number of generators could vary wildly depending on
the orientation of the edges. Furthermore, $\mathcal{D}_{1}$ has two
generators having the same supports with different exponents which never
happens in the simple edge graph case.

######  Acknowledgment.

The authors are grateful to the software system Macaulay2 [8], for serving as
an excellent source of inspiration.

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