2 Tropical ideals

In this section we introduce tropical ideals, together with several examples. We first recall some basics of valuated matroids.

Throughout this paper we take $(R,\oplus ,\odot )$ to be a semifield, with the following extra properties. For compatibility with tropical notation we write $\infty$ for the additive identity of $R$ , and $R^{\ast }$ for $R\setminus \{\infty \}$ . We require that $(R^{\ast },\odot )$ is a totally ordered group, and that the addition $\oplus$ satisfies $a\oplus b=\min (a,b)$ . Under these hypotheses, $R$ is sometimes called a valuative semifield. We extend the total ordering on the multiplicative group to all of $R$ by making $\infty$ the largest element.

The main example for us will be the tropical semiring (or min-plus algebra) $\overline{\mathbb{R}}$ :

$$\begin{eqnarray}\overline{\mathbb{R}}:=(\mathbb{R}\cup \{\infty \},\oplus ,\odot )\quad \text{where }\oplus :=\min \text{ and }\odot :=+.\end{eqnarray}$$

Here the ordering on the multiplicative group $\mathbb{R}$ is the standard one. Another important example is the Boolean subsemiring $\mathbb{B}$ of $\overline{\mathbb{R}}$ :

$$\begin{eqnarray}\mathbb{B}:=(\{0,\infty \},\oplus ,\odot ).\end{eqnarray}$$

An ideal in a semiring $R$ is a nonempty subset of $R$ closed under addition and under multiplication by elements of $R$ .

We denote by $R[x_{0},\ldots ,x_{n}]$ the semiring of polynomials in the variables $x_{0},\ldots ,x_{n}$ with coefficients in $R$ . Note that in the case that $R=\overline{\mathbb{R}}$ , elements of $\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ are regarded as polynomials and not functions; for example, the polynomials $f(x)=x^{2}\,\oplus \,0$ and $g(x)=x^{2}\,\oplus \,1\odot x\,\oplus \,0$ are distinct, even though $f(w)=g(w)$ for all $w\in \mathbb{R}$ . The support of a polynomial $f=\bigoplus a_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}$ is

$$\begin{eqnarray}\operatorname{supp}(f):=\{\mathbf{u}\in \mathbb{N}^{n+1}:a_{\boldsymbol{ u}}\neq \infty \}.\end{eqnarray}$$

We call $a_{\mathbf{u}}$ the coefficient in $f$ of the monomial $\mathbf{x}^{\mathbf{u}}$ .

2.1 Valuated matroids

Valuated matroids are a generalization of the notion of matroids, introduced by Dress and Wenzel in [Reference Dress and WenzelDW92]. We recall some of the necessary background on valuated matroids and tropical linear spaces; for basics of standard matroids, see, for example, [Reference OxleyOxl92].

Let $E$ be a finite set, and let $r\in \mathbb{N}$ . Denote by $\binom{E}{r}$ the collection of subsets of $E$ of size $r$ . A valuated matroid on the ground set $E$ with values in the semifield $R$ is a pair ${\mathcal{M}}=(E,p)$ where $p:\binom{E}{r}\rightarrow R$ satisfies the following properties.

1. – There exists $B\in \binom{E}{r}$ such that $p(B)\neq \infty$ .

2. – Valuated basis exchange axiom: for every $A,B\in \binom{E}{r}$ and every $a\in A\setminus B$ there exists $b\in B\setminus A$ with

   (2) $$\begin{eqnarray}p(A)\odot p(B)\geqslant p(A\cup b-a)\odot p(B\cup a-b).\end{eqnarray}$$

In the case that $R=\overline{\mathbb{R}}$ , the valuated basis exchange axiom is equivalent to the tropical Plücker relations; see, for instance, [Reference Maclagan and SturmfelsMS15, §4.4]. If ${\mathcal{M}}=(E,p)$ is a valuated matroid, its support $\{B\in \binom{E}{r}:p(B)\neq \infty \}$ is the collection of bases of a rank- $r$ matroid on the ground set $E$ , called the underlying matroid $\text{}\underline{{\mathcal{M}}}$ of ${\mathcal{M}}$ . The function $p$ is called the basis valuation function of ${\mathcal{M}}$ . We consider the basis valuation functions $p$ and $\unicode[STIX]{x1D706}\odot p$ with $\unicode[STIX]{x1D706}\in R^{\ast }$ to be the same valuated matroid.

As for ordinary matroids, valuated matroids have several different ‘cryptomorphic’ definitions, some of which we now recall. For more information, see [Reference Murota and TamuraMT01].

Let ${\mathcal{M}}$ be a valuated matroid on the ground set $E$ with basis valuation function $p:\binom{E}{r}\rightarrow R$ . Given a basis $B$ of $\text{}\underline{{\mathcal{M}}}$ and an element $e\in E\setminus B$ , the (valuated) fundamental circuit $H(B,e)$ of ${\mathcal{M}}$ is the element of the $R$ -semimodule $R^{E}$ whose coordinates are given by

(3) $$\begin{eqnarray}H(B,e)_{e^{\prime }}:=p(B\cup e-e^{\prime })/p(B)\quad \text{for any }e^{\prime }\in E,\end{eqnarray}$$

where $/$ denotes division in the semifield $R$ (subtraction in the case of $\overline{\mathbb{R}}$ ), $p(B^{\prime })=\infty$ if $|B^{\prime }|>r$ , and $\infty /\unicode[STIX]{x1D706}=\infty$ for any $\unicode[STIX]{x1D706}\in R^{\ast }$ . A (valuated) circuit of ${\mathcal{M}}$ is any vector in $R^{E}$ of the form $\unicode[STIX]{x1D706}\odot H(B,e)$ , where $B$ is a basis of $\text{}\underline{{\mathcal{M}}}$ , $e\in E\setminus B$ , and $\unicode[STIX]{x1D706}\in R^{\ast }$ . We denote by ${\mathcal{C}}({\mathcal{M}})$ the collection of all circuits of ${\mathcal{M}}$ . For any $H\in R^{E}$ , its support is defined as

$$\begin{eqnarray}\operatorname{supp}(H):=\{e\in E:H_{e}\neq \infty \}.\end{eqnarray}$$

The set of supports of the circuits of ${\mathcal{M}}$ is equal to the set of circuits of the underlying matroid $\text{}\underline{{\mathcal{M}}}$ . Furthermore, if two circuits $G$ and $H$ of ${\mathcal{M}}$ have the same support, then there exists $\unicode[STIX]{x1D706}\in R^{\ast }$ such that $G=\unicode[STIX]{x1D706}\odot H$  [Reference Murota and TamuraMT01, Theorem 3.1(VC $3_{e}$ )].

Collections of circuits of valuated matroids can be intrinsically characterized by axioms that generalize the classical circuit axioms for matroids; see [Reference Murota and TamuraMT01, Theorem 3.1]. The most important one is the following elimination property.

1. – Circuit elimination axiom: For any $G,H\in {\mathcal{C}}({\mathcal{M}})$ and any $e,e^{\prime }\in E$ such that $G_{e}=H_{e}\neq \infty$ and $G_{e^{\prime }}<H_{e^{\prime }}$ , there exists $F\in {\mathcal{C}}({\mathcal{M}})$ satisfying $F_{e}=\infty$ , $F_{e^{\prime }}=G_{e^{\prime }}$ , and $F\geqslant G\oplus H$ .

Here $F\geqslant F^{\prime }$ if $F_{e}\geqslant F_{e}^{\prime }$ for all $e$ , and $(G\oplus H)_{e}=G_{e}\oplus H_{e}$ .

We also use the vector formulation for valuated matroids, which generalizes the notion of cycles for matroids. A cycle of a matroid is a union of circuits. A vector of a valuated matroid is an element of the subsemimodule of $R^{E}$ generated by the valuated circuits. More explicitly, the set of vectors of ${\mathcal{M}}$ is

$$\begin{eqnarray}{\mathcal{V}}({\mathcal{M}}):=\biggl\{\bigoplus _{H\in {\mathcal{C}}({\mathcal{M}})}\,\unicode[STIX]{x1D706}_{H}\odot H:\unicode[STIX]{x1D706}_{H}\in R\text{ for all }H\biggr\}.\end{eqnarray}$$

Vectors of a valuated matroid can also be characterized by axioms; see [Reference Murota and TamuraMT01, Theorem 3.4]. In fact, a subset ${\mathcal{V}}\subseteq R^{E}$ is the set of vectors of a valuated matroid if and only if it is a subsemimodule of $R^{E}$ satisfying the following property.

1. – Vector elimination axiom: For any $G,H\in {\mathcal{V}}$ and any $e\in E$ such that $G_{e}=H_{e}\neq \infty$ , there exists $F\in {\mathcal{V}}$ satisfying $F_{e}=\infty$ , $F\geqslant G\oplus H$ , and $F_{e^{\prime }}=G_{e^{\prime }}\oplus H_{e^{\prime }}$ for all $e^{\prime }\in E$ such that $G_{e^{\prime }}\neq H_{e^{\prime }}$ .

In the case that $R=\mathbb{B}$ , valuated matroids are simply matroids, as there is no additional information encoded in the valuation. In this case, valuated circuits and vectors are in one-to-one correspondence with circuits and cycles, respectively, of the underlying matroid $\text{}\underline{{\mathcal{M}}}$ .

In the case that $R=\overline{\mathbb{R}}$ , the set of vectors of a valuated matroid is also called a tropical linear space in the tropical literature. In the terminology used in [Reference Maclagan and SturmfelsMS15, §4.4], if $p$ is the basis valuation function of a rank- $r$ valuated matroid ${\mathcal{M}}$ , then ${\mathcal{V}}({\mathcal{M}})$ is the tropical linear space $L_{p^{\bot }}$ , where $p^{\bot }$ is the dual tropical Plücker vector given by $p^{\bot }(B):=p(E\setminus B)$ .

2.2 Tropical ideals

We now introduce the main object of study of this paper. Let $\operatorname{Mon}_{d}$ be the set of monomials of degree $d$ in the variables $x_{0},\ldots ,x_{n}$ . We will identify elements of $R^{\operatorname{Mon}_{d}}$ with homogeneous polynomials of degree $d$ in $R[x_{0},\ldots ,x_{n}]$ . In this way, if ${\mathcal{M}}$ is a valuated matroid on the ground set $\operatorname{Mon}_{d}$ , circuits and vectors of ${\mathcal{M}}$ can be thought of as homogeneous polynomials in $R[x_{0},\ldots ,x_{n}]$ of degree $d$ .

Definition 2.1 (Homogeneous tropical ideals).

A homogeneous tropical ideal is a homogeneous ideal $I\subseteq R[x_{0},\ldots ,x_{n}]$ such that for each $d\geqslant 0$ the degree- $d$ part $I_{d}$ is the collection of vectors of a valuated matroid ${\mathcal{M}}_{d}$ on $\operatorname{Mon}_{d}$ . If $I\subseteq R[x_{0},\ldots ,x_{n}]$ is a homogeneous tropical ideal, we will denote by ${\mathcal{M}}_{d}(I)$ the valuated matroid such that $I_{d}={\mathcal{V}}({\mathcal{M}}_{d}(I))$ .

This definition is consistent with Definition 1.1 in the introduction, in view of the characterization of vectors of a valuated matroid in terms of the vector elimination axiom.

Not all homogeneous ideals in $R[x_{0},\ldots ,x_{n}]$ are tropical ideals. As an example, consider the ideal $I$ in $\overline{\mathbb{R}}[x,y]$ generated by $x\oplus y$ . The degree-2 part of this ideal is the $\overline{\mathbb{R}}$ -semimodule generated by $x^{2}\oplus xy$ and $xy\oplus y^{2}$ . This is not the set of vectors of a valuated matroid on $\operatorname{Mon}_{2}=\{x^{2},xy,y^{2}\}$ , as the polynomial $x^{2}\oplus y^{2}$ would be required to be in $I$ by the vector elimination axiom applied to the two generators.

Homogeneous tropical ideals in the sense of Definition 2.1 will define subschemes of tropical projective space. We elaborate on this definition with some examples.

Example 2.2 (Realizable tropical ideals).

Let $K$ be a field equipped with a valuation $\operatorname{val}:K\,\rightarrow \,R$ . Any polynomial $g\in K[x_{0},\ldots ,x_{n}]$ gives rise to a ‘tropical’ polynomial $\operatorname{trop}(g)\in R[x_{0},\ldots ,x_{n}]$ by interpreting all operations tropically and replacing coefficients by their valuations: if $g=\sum c_{\mathbf{u}}\cdot \mathbf{x}^{\mathbf{u}}$ , then $\operatorname{trop}(g):=\bigoplus \operatorname{val}(c_{\mathbf{u}})\odot \mathbf{x}^{\mathbf{u}}$ . For any ideal $J\subseteq K[x_{0},\ldots ,x_{n}]$ , its tropicalization is the ideal

$$\begin{eqnarray}\operatorname{trop}(J):=\langle \operatorname{trop}(g):g\in J\rangle \subseteq R[x_{0},\ldots ,x_{n}].\end{eqnarray}$$

If $f$ is a polynomial of minimal support in $\operatorname{trop}(J)$ , there exists $g\in J$ and $\unicode[STIX]{x1D706}\in R^{\ast }$ such that $f=\unicode[STIX]{x1D706}\odot \operatorname{trop}(g)$ . Indeed, if $f=\sum \unicode[STIX]{x1D706}_{i}\odot \mathbf{x}^{\mathbf{u}_{i}}\odot \operatorname{trop}(g_{i})$ with $g_{i}\in J$ for all $i$ , then $f=\sum \unicode[STIX]{x1D706}_{i}\odot \operatorname{trop}(\mathbf{x}^{\mathbf{u}_{i}}g_{i})$ . Since there is no cancelation in $R$ , we have $\operatorname{supp}(\mathbf{x}^{\mathbf{u}_{i}}g_{i})\subseteq \operatorname{supp}(f)$ for all $i$ , and thus $\operatorname{supp}(\mathbf{x}^{\mathbf{u}_{i}}g_{i})=\operatorname{supp}(f)$ for all $i$ . As all the $\mathbf{x}^{\mathbf{u}_{i}}g_{i}$ are of minimal support in $J$ , this implies that all of them are scalar multiples of each other, and thus $f=\unicode[STIX]{x1D706}\odot \operatorname{trop}(g)$ for some $g\in J$ .

It follows that if $J\subseteq K[x_{0},\ldots ,x_{n}]$ is a homogeneous ideal, for any $d\geqslant 0$ the degree- $d$ polynomials in $\operatorname{trop}(J)$ of minimal support satisfy the circuit elimination axiom of valuated matroids. The other circuit axioms are immediate, so $\operatorname{trop}(J)$ is a homogeneous tropical ideal. A tropical ideal arising in this way is called realizable (over the field $K$ ).

Remark 2.3. If $J\subseteq K[x_{0},\ldots ,x_{n}]$ is a homogeneous ideal, the degree- $d$ part $\operatorname{trop}(J)_{d}$ of the homogeneous tropical ideal $\operatorname{trop}(J)$ is the $R$ -semimodule generated by the polynomials $\operatorname{trop}(g)$ with $g\in J_{d}$ . Moreover, if the residue field of $K$ is infinite, then every polynomial in $\operatorname{trop}(J)_{d}$ is a scalar multiple of some $\operatorname{trop}(g)$ with $g\in J_{d}$ , as $\operatorname{trop}(g)\oplus \operatorname{trop}(h)=\operatorname{trop}(g+\unicode[STIX]{x1D6FC}h)$ for a sufficiently general $\unicode[STIX]{x1D6FC}\in K$ with $\operatorname{val}(\unicode[STIX]{x1D6FC})=0$ . If, in addition, the value group $\unicode[STIX]{x1D6E4}:=\operatorname{im}\operatorname{val}$ of $K$ equals all of $R$ , then every polynomial in $\operatorname{trop}(J)_{d}$ is of the form $\operatorname{trop}(g)$ for $g\in J_{d}$ .

When the residue field is finite, however, it is possible for the underlying matroid $\text{}\underline{{\mathcal{M}}_{d}(\operatorname{trop}(J))}$ to have some cycles that are not the support of any polynomial in $J$ . For example, consider $K=\mathbb{Z}/2\mathbb{Z}$ equipped with the trivial valuation $\operatorname{val}:K\rightarrow \mathbb{B}$ , and the ideal $J:=\langle x+y\rangle \subseteq K[x,y]$ . In this case there is no polynomial in $J_{2}$ having support $D:=\{x^{2},xy,y^{2}\}$ , even though the tropical polynomial $x^{2}\oplus xy\oplus y^{2}$ is in $\operatorname{trop}(J)_{2}$ and $D$ is a cycle of $\text{}\underline{{\mathcal{M}}_{2}(\operatorname{trop}(J))}$ .

A special class of realizable homogeneous tropical ideals consists of the tropical equivalent of the homogeneous ideal of a point in projective space.

Example 2.4 (Homogeneous tropical ideal of a point).

Fix $\mathbf{a}=(a_{0},a_{1},\ldots ,a_{n})\in \operatorname{trop}(\mathbb{P}^{n})=(\overline{\mathbb{R}}^{n+1}\setminus \{(\infty ,\ldots ,\infty )\})/\mathbb{R}(1,\ldots ,1)$ . Let $I_{\mathbf{a}}$ be the ideal generated by all homogeneous polynomials $f\in \overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ for which $\mathbf{a}\in V(f)$ , so $f(\mathbf{a})=\infty$ or the minimum in $f(\mathbf{a})$ is achieved at least twice. We claim that $I_{\mathbf{a}}$ is a tropical ideal. In addition, if $K$ is a valued field with $a_{i}$ in the image of the valuation for each $0\leqslant i\leqslant n$ , then $I_{\mathbf{a}}$ is the tropicalization of any ideal $J_{\unicode[STIX]{x1D6FC}}$ of a point $\unicode[STIX]{x1D6FC}\in \mathbb{P}_{K}^{n}$ with $\operatorname{val}(\unicode[STIX]{x1D6FC}_{i})=a_{i}$ .

To prove the first claim, we first note that $I_{\mathbf{a}}$ is generated as an $\overline{\mathbb{R}}$ -semimodule by the set ${\mathcal{P}}$ of polynomials of the form $(\mathbf{a}\cdot \mathbf{v})\odot \mathbf{x}^{\mathbf{u}}\oplus (\mathbf{a}\cdot \mathbf{u})\odot \mathbf{x}^{\mathbf{v}}$ with $\deg (\mathbf{x}^{\mathbf{u}})=\deg (\mathbf{x}^{\mathbf{v}})$ and $\mathbf{u}\neq \mathbf{v}$ , where by convention we take $\infty$ times $0$ equal to $0$ when some of the $a_{i}$ are equal to $\infty$ . Indeed, all these binomials (and monomials, in the case some of the $a_{i}$ equal $\infty$ ) are contained in $I_{\mathbf{a}}$ . Suppose that $f=\bigoplus c_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}\in I_{\mathbf{a}}$ is outside the ideal generated by ${\mathcal{P}}$ . We may assume that $f$ has been chosen to have as few terms as possible. We may also assume that $f(\mathbf{a})<\infty$ , as otherwise all terms of $f$ lie in the ideal generated by ${\mathcal{P}}$ . Fix $c_{\mathbf{v}}\odot \mathbf{x}^{\mathbf{v}}$ to be a term of $f$ such that $c_{\mathbf{v}}+\mathbf{a}\cdot \mathbf{v}>f(\mathbf{a})$ if one exists, or any term of $f$ otherwise. Take $\mathbf{v}^{\prime }\neq \mathbf{v}$ with $c_{\mathbf{v}^{\prime }}+\mathbf{a}\cdot \mathbf{v}^{\prime }=f(\mathbf{a})$ , and set $g=\bigoplus _{\mathbf{u}\neq \mathbf{v}}c_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}$ . Then $f=g\oplus (c_{\mathbf{v}}-\mathbf{a}\cdot \mathbf{v}^{\prime })\odot ((\mathbf{a}\cdot \mathbf{v}^{\prime })\odot \mathbf{x}^{\mathbf{v}}\oplus (\mathbf{a}\cdot \mathbf{v})\odot \mathbf{x}^{\mathbf{v}^{\prime }})$ . In the case that the minimum in $f(\mathbf{a})$ is not achieved at $\mathbf{x}^{\mathbf{v}}$ , the minimum in $g(\mathbf{a})$ is still achieved at least twice. If the minimum in $f(\mathbf{a})$ is achieved at all its terms, $f$ must have at least three terms, as otherwise it would be a scalar multiple of one of the polynomials in ${\mathcal{P}}$ . The minimum in $g(\mathbf{a})$ is thus still achieved twice. Furthermore, $g$ has fewer terms than $f$ , so by assumption $g$ lies in the $\overline{\mathbb{R}}$ -semimodule generated by ${\mathcal{P}}$ , and thus so does $f$ . This proves that $I_{\mathbf{a}}$ is generated by ${\mathcal{P}}$ . In addition, the degree- $d$ polynomials of minimal support in ${\mathcal{P}}$ satisfy the valuated circuit elimination axiom, which implies that $(I_{\mathbf{a}})_{d}$ is the set of vectors of a valuated matroid [Reference Murota and TamuraMT01, Theorem 3.4], and thus $I_{\mathbf{a}}$ is a tropical ideal.

Suppose now that $\unicode[STIX]{x1D6FC}\in \mathbb{P}_{K}^{n}$ satisfies $\operatorname{val}(\unicode[STIX]{x1D6FC})=\mathbf{a}$ . The homogeneous ideal $J_{\unicode[STIX]{x1D6FC}}\subseteq K[x_{0},\ldots ,x_{n}]$ of the point $\unicode[STIX]{x1D6FC}$ contains the binomials $\unicode[STIX]{x1D6FC}^{\mathbf{v}}\mathbf{x}^{\mathbf{u}}-\unicode[STIX]{x1D6FC}^{\mathbf{u}}\mathbf{x}^{\mathbf{v}}$ for all pairs $\mathbf{x}^{\mathbf{u}},\mathbf{x}^{\mathbf{v}}$ with $\deg (\mathbf{x}^{\mathbf{u}})=\deg (\mathbf{x}^{\mathbf{v}})$ , so $I_{\mathbf{a}}\subseteq \operatorname{trop}(J_{\unicode[STIX]{x1D6FC}})$ . If the inclusion were proper, there would be $h\in \operatorname{trop}(J_{\unicode[STIX]{x1D6FC}})$ with $h\notin I_{\mathbf{a}}$ . Write $h=\sum h_{i}\odot \operatorname{trop}(g_{i})$ , where $g_{i}\in J_{\unicode[STIX]{x1D6FC}}$ . Since $\mathbf{a}=\operatorname{val}(\unicode[STIX]{x1D6FC})\in \operatorname{trop}(V(g_{i}))=V(\operatorname{trop}(g_{i}))$ , this contradicts that $\mathbf{a}\not \in V(h)$ .

One can think of a homogeneous tropical ideal as the ‘tower’ of valuated matroids that determine its various homogeneous parts, as described in the following definition.

Definition 2.5 (Compatible valuated matroids).

Let ${\mathcal{S}}=({\mathcal{M}}_{d})_{d\geqslant 0}$ be a sequence of valuated matroids with values in the semiring $R$ , where the ground set of ${\mathcal{M}}_{d}$ is the set of monomials $\operatorname{Mon}_{d}$ . The sequence ${\mathcal{S}}$ is called a compatible sequence if the $R$ -subsemimodule $I$ of $R[x_{0},\ldots ,x_{n}]$ generated by $\{{\mathcal{V}}({\mathcal{M}}_{d}):d\geqslant 0\}$ is a homogeneous ideal. In other words, ${\mathcal{S}}$ is compatible if for any $d\geqslant 0$ and any variable $x_{i}$ we have

$$\begin{eqnarray}x_{i}\odot {\mathcal{V}}({\mathcal{M}}_{d}):=\{x_{i}\odot H:H\in {\mathcal{V}}({\mathcal{M}}_{d})\}\subseteq {\mathcal{V}}({\mathcal{M}}_{d+1}),\end{eqnarray}$$

or equivalently for any circuit $G\in {\mathcal{C}}({\mathcal{M}}_{d})$ we have that $x_{i}\odot G$ is a sum of circuits of ${\mathcal{M}}_{d+1}$ .

Compatibility of valuated matroids can be simply described in terms of their basis valuation functions.

Proposition 2.6 (Compatibility in terms of basis valuations).

Let ${\mathcal{S}}=({\mathcal{M}}_{d})_{d\geqslant 0}$ be a sequence of valuated matroids, where ${\mathcal{M}}_{d}=(\operatorname{Mon}_{d},p_{d})$ has rank $r_{d}$ . The sequence ${\mathcal{S}}$ is a compatible sequence if and only if for any $d\geqslant 0$ , any variable $x_{i}$ , any $U\subseteq \operatorname{Mon}_{d}$ of size $r_{d}+1$ , and any $V\subseteq \operatorname{Mon}_{d+1}$ of size $r_{d+1}-1$ , we have that

$$\begin{eqnarray}\min _{\mathbf{x}^{\mathbf{u}}\in x_{i}U\setminus V}p_{d}(U-\mathbf{x}^{\mathbf{u}}/x_{i})\odot p_{d+1}(V\cup \mathbf{x}^{\mathbf{u}})\text{ is attained at least twice}.\end{eqnarray}$$

Proof. For any $d\geqslant 0$ and any variable $x_{i}$ , the set $x_{i}\odot {\mathcal{V}}({\mathcal{M}}_{d})$ is the collection of vectors of a valuated matroid on the ground set $\operatorname{Mon}_{d+1}$ . Indeed, if we let $\operatorname{Mon}_{d}^{\hat{i} }$ be the set of monomials in $\operatorname{Mon}_{d+1}$ not divisible by $x_{i}$ , the collection $x_{i}\odot {\mathcal{V}}({\mathcal{M}}_{d})$ is the set of vectors of the valuated matroid ${\mathcal{M}}_{d,i}$ on $\operatorname{Mon}_{d+1}$ with basis valuation function

$$\begin{eqnarray}p_{d,i}(B)=\left\{\begin{array}{@{}ll@{}}p_{d}((B\setminus \operatorname{Mon}_{d}^{\hat{i} })/x_{i})\quad & \text{if }B\supseteq \operatorname{Mon}_{d}^{\hat{i} },\\ \infty \quad & \text{otherwise.}\end{array}\right.\end{eqnarray}$$

The sequence ${\mathcal{S}}$ is compatible if and only if, for every $d\geqslant 0$ and any variable $x_{i}$ , there is a containment of tropical linear spaces ${\mathcal{V}}({\mathcal{M}}_{d,i})\subseteq {\mathcal{V}}({\mathcal{M}}_{d+1})$ , or dually ${\mathcal{V}}({\mathcal{M}}_{d,i})^{\bot }\supseteq {\mathcal{V}}({\mathcal{M}}_{d+1})^{\bot }$ . By [Reference HaqueHaq12, Theorem 1], this is equivalent to the condition that, for any $V,V^{\prime }\subseteq \operatorname{Mon}_{d+1}$ with $|V|=|\!\operatorname{Mon}_{d}^{\hat{i} }|+r_{d}+1$ and $|V^{\prime }|=r_{d+1}-1$ ,

$$\begin{eqnarray}\min _{\mathbf{v}\in V\setminus V^{\prime }}p_{d,i}(V-\mathbf{v})\odot p_{d+1}(V^{\prime }\cup \mathbf{v})\text{ is attained at least twice},\end{eqnarray}$$

which reduces to the desired condition. ◻

The next example shows that tropical ideals carry strictly more information than their varieties.

Figure 1. Two tropical irreducible decompositions.

Example 2.7 (Two tropical ideals with the same variety).

Let $K$ be the field $\mathbb{C}$ equipped with the trivial valuation $\operatorname{val}:\mathbb{C}\rightarrow \overline{\mathbb{R}}$ . Consider the principal ideals $J=\langle (x+y+z)(xy+xz+yz)\rangle$ and $J^{\prime }=\langle (x+y)(x+z)(y+z)\rangle$ in $\mathbb{C}[x,y,z]$ . Their tropicalizations $I:=\operatorname{trop}(J)$ and $I^{\prime }:=\operatorname{trop}(J^{\prime })$ are homogeneous tropical ideals in $\overline{\mathbb{R}}[x,y,z]$ with the same variety. This is shown in Figure 1, together with the two associated tropical irreducible decompositions corresponding to tropicalizing the factors in the generators of $J$ and $J^{\prime }$ . However, even though the tropical ideals $I$ and $I^{\prime }$ coincide up to degree 3, they are distinct. For instance, the degree-4 polynomial

$$\begin{eqnarray}f:=x^{3}y\oplus x^{3}z\oplus xy^{3}\oplus y^{3}z\oplus xz^{3}\oplus yz^{3}\oplus xy^{2}z\oplus xyz^{2}\oplus y^{2}z^{2}\end{eqnarray}$$

is equal to $\operatorname{trop}((x+y)(x+z)(y+z)(x-y-z))\in I^{\prime }$ , but $f$ is not in $I$ : a simple computation shows that no polynomial of the form $(x+y+z)(xy+xz+yz)(ax+by+cz)$ can have support equal to the support of $f$ .

We finish this section with an example of a tropical ideal that is not realizable over any field  $K$ .

Example 2.8 (A non-realizable tropical ideal).

For any $n\geqslant 2$ , we give an example of a homogeneous tropical ideal in $\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ that is not realizable over any valued field. Its variety is, however, the standard tropical line in tropical projective space; see Example 5.15.

For $d\geqslant 0$ , let ${\mathcal{M}}_{d}$ be the rank-( $d+1$ ) valuated matroid on the ground set $\operatorname{Mon}_{d}$ whose basis valuation function $p_{d}:\binom{\operatorname{Mon}_{d}}{d+1}\rightarrow \overline{\mathbb{R}}$ is given by

$$\begin{eqnarray}p_{d}(B):=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if for any }k\leqslant d\text{ and any }\mathbf{x}^{\mathbf{v}}\in \operatorname{Mon}_{k}\text{ we have }\\ \quad & \quad |\{\mathbf{x}^{\mathbf{u}}\in B:\mathbf{x}^{\mathbf{v}}\text{ divides }\mathbf{x}^{\mathbf{u}}\}|\leqslant d-k+1,\\ \infty \quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$

Geometrically, if we think of $\operatorname{Mon}_{d}$ as the set of lattice points inside a simplex of size $d$ , the function $p_{d}$ assigns the value $0$ precisely to those $(d+1)$ -subsets $B$ of $\operatorname{Mon}_{d}$ such that, for any $k\leqslant d$ , the subset $B$ contains at most $d-k+1$ monomials from any simplex in $\operatorname{Mon}_{d}$ of size $d-k$ . A proof that the underlying matroids $\text{}\underline{{\mathcal{M}}_{d}}$ are indeed matroids (and thus the ${\mathcal{M}}_{d}$ are valuated matroids) can be found in [Reference Ardila and BilleyAB07, Theorem 4.1], where these matroids are studied in connection to flag arrangements.

The circuits of ${\mathcal{M}}_{d}$ are the tropical polynomials of the form $H=\unicode[STIX]{x1D706}\odot \bigoplus _{\mathbf{u}\in C}\mathbf{x}^{\mathbf{u}}$ with $\unicode[STIX]{x1D706}\in \mathbb{R}$ and $C$ an inclusion-minimal subset of $\operatorname{Mon}_{d}$ satisfying $|C|>d-\deg (\gcd (C))+1$ . This description shows that, if $H$ is a circuit of ${\mathcal{M}}_{d}$ and $x_{i}$ is any variable, then $x_{i}\odot H$ is a circuit of ${\mathcal{M}}_{d+1}$ . Thus $({\mathcal{M}}_{d})_{d\geqslant 0}$ is a compatible sequence of valuated matroids, and so the $\overline{\mathbb{R}}$ -semimodule $I$ generated by all ${\mathcal{V}}({\mathcal{M}}_{d})$ is a homogeneous tropical ideal.

To show that $I$ is a non-realizable tropical ideal, note that the tropical polynomial $f=x_{0}\oplus x_{1}\oplus x_{2}$ is a circuit of ${\mathcal{M}}_{1}$ , so in particular $f\in I$ . If $I$ is realizable, then $I=\operatorname{trop}(J)$ for some ideal $J\subseteq K[x_{0},\ldots ,x_{n}]$ , and so, since $f$ is a circuit, there exists some $g\in J$ such that $f=\operatorname{trop}(g)$ . After a suitable scaling of the variables, we may assume that $g=x_{0}+x_{1}+x_{2}$ . The polynomial

$$\begin{eqnarray}h:=g\cdot (x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{0}x_{1}-x_{0}x_{2}-x_{1}x_{2})=x_{0}^{3}+x_{1}^{3}+x_{2}^{3}-3x_{0}x_{1}x_{2}\end{eqnarray}$$

is then a polynomial in $J$ , and thus $\operatorname{trop}(h)$ lies in $I$ . However, this contradicts the fact that $\operatorname{supp}(\operatorname{trop}(h))\subseteq \{x_{0}^{3},x_{1}^{3},x_{2}^{3},x_{0}x_{1}x_{2}\}$ is an independent set in the underlying matroid $\text{}\underline{{\mathcal{M}}_{3}}$ .

3 Gröbner theory for tropical ideals

In this section we develop a Gröbner theory for homogeneous tropical ideals in $\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ and $\mathbb{B}[x_{0},\ldots ,x_{n}]$ , and use it to prove some basic properties of tropical ideals with coefficients in a more general semiring $R$ . These include the eventual polynomiality of their Hilbert functions, and the fact that tropical ideals satisfy the ascending chain condition.

We start by defining initial ideals of homogeneous ideals in $\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ with respect to a weight vector $\mathbf{w}\in \overline{\mathbb{R}}^{n+1}$ .

Definition 3.1 (Initial ideals).

Let $\mathbf{w}\in \overline{\mathbb{R}}^{n+1}$ with $\mathbf{w}\neq (\infty ,\ldots ,\infty )$ . The initial term of a tropical polynomial $f=\bigoplus a_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}\in \overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ with respect to $\mathbf{w}$ is the tropical polynomial in $\mathbb{B}[x_{0},\ldots ,x_{n}]$ given by

$$\begin{eqnarray}\operatorname{in}_{\mathbf{w}}(f):=\left\{\begin{array}{@{}ll@{}}\displaystyle \underset{a_{\mathbf{u}}+\mathbf{w}\cdot \mathbf{u}=f(\mathbf{w})}{\bigoplus }\mathbf{x}^{\mathbf{u}}\quad & \text{if }f(\mathbf{w})<\infty ,\\ \qquad \infty \quad & \text{otherwise},\end{array}\right.\quad \in \mathbb{B}[x_{0},\ldots ,x_{n}].\end{eqnarray}$$

In computing the dot product $\mathbf{w}\cdot \mathbf{u}$ we follow the convention that $\infty$ times $a$ is $\infty$ for $a\neq 0$ , and $\infty$ times $0$ is $0$ , so $\mathbf{w}\cdot \mathbf{u}=\mathbf{w}^{\mathbf{u}}$ . The initial ideal $\operatorname{in}_{\mathbf{w}}(I)$ of a homogeneous ideal $I$ in $\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ is the homogeneous ideal

$$\begin{eqnarray}\operatorname{in}_{\mathbf{w}}(I):=\langle \operatorname{in}_{\mathbf{w}}(f):f\in I\rangle \subseteq \mathbb{B}[x_{0},\ldots ,x_{n}].\end{eqnarray}$$

If $I$ is a homogeneous ideal in $\mathbb{B}[x_{0},\ldots ,x_{n}]$ , then we define its initial ideal $\operatorname{in}_{\mathbf{w}}(I)$ as the initial ideal of $I\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ , using the inclusion of $\mathbb{B}$ into $\overline{\mathbb{R}}$ .

Note that $\operatorname{in}_{\mathbf{w}}(a\odot f\oplus b\odot g)=\operatorname{in}_{\mathbf{w}}(f)\oplus \operatorname{in}_{\mathbf{w}}(g)$ for $f,g\in I$ and $a,b\in \overline{\mathbb{R}}$ such that $a\odot f(\mathbf{w})=b\odot g(\mathbf{w})<\infty$ , so the set of initial forms of a graded piece $I_{d}$ of $I$ is a semimodule over $\mathbb{B}$ . As, in addition, $\operatorname{in}_{\mathbf{w}}(x_{i}\odot f)=x_{i}\odot \operatorname{in}_{\mathbf{w}}(f)$ , we have $\operatorname{in}_{\mathbf{w}}(I_{d})=(\operatorname{in}_{\mathbf{w}}I)_{d}$ . Also, since $I$ is a homogeneous ideal, $\operatorname{in}_{\mathbf{w}}(I)=\operatorname{in}_{\mathbf{w}+\unicode[STIX]{x1D706}\mathbf{1}}(I)$ for $\mathbf{1}=(1,1,\ldots ,1)$ , so we may regard $\mathbf{w}$ as an element of $\operatorname{trop}(\mathbb{P}^{n})$ .

An important result in this section is that initial ideals of tropical ideals are also tropical ideals. This will follow from the following key fact about valuated matroids.

We first extend the notion of initial term to vectors of any valuated matroid over $\overline{\mathbb{R}}$ . If $E$ is any finite set, $H\in \overline{\mathbb{R}}^{E}$ , and $\mathbf{w}\in \overline{\mathbb{R}}^{E}$ with $\mathbf{w}\neq (\infty ,\ldots ,\infty )$ , the initial term of $H$ with respect to $\mathbf{w}$ is the subset of $E$ ,

$$\begin{eqnarray}\operatorname{in}_{\mathbf{w}}H:=\{e\in E:H_{e}+w_{e}\text{ is minimal among all }e\in E\}\end{eqnarray}$$

if $\min (H_{e}+w_{e})<\infty$ , and $\operatorname{in}_{\mathbf{w}}H:=\emptyset$ otherwise.

Lemma 3.3. Let ${\mathcal{M}}=(E,p)$ be a rank- $r$ valuated matroid, where $p:\binom{E}{r}\rightarrow \overline{\mathbb{R}}$ is its basis valuation function, and let $\mathbf{w}=(w_{e})_{e\in E}\in \mathbb{R}^{E}$ . Then

$$\begin{eqnarray}\operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}}):=\biggl\{B\in \binom{E}{r}:p(B)-\mathop{\sum }_{e\in B}w_{e}\text{ is minimal among all }B\in \binom{E}{r}\biggr\}\end{eqnarray}$$

is the collection of bases of an (ordinary) matroid $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ of rank $r$ on the ground set $E$ . Its circuits are the elements of $\operatorname{in}_{\mathbf{w}}{\mathcal{C}}({\mathcal{M}}):=\{\operatorname{in}_{\mathbf{w}}H:H\in {\mathcal{C}}({\mathcal{M}})\}$ that are minimal with respect to inclusion, and its set of cycles is

$$\begin{eqnarray}\operatorname{in}_{\mathbf{w}}{\mathcal{V}}({\mathcal{M}}):=\{\operatorname{in}_{\mathbf{w}}H:H\in {\mathcal{V}}({\mathcal{M}})\}.\end{eqnarray}$$

Proof. For any $B\in \binom{E}{r}$ set

$$\begin{eqnarray}p_{\mathbf{w}}(B):=p(B)-\mathop{\sum }_{e\in B}w_{e}.\end{eqnarray}$$

To prove that $\operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})$ satisfies the basis exchange axiom, fix $A,B\in \binom{E}{r}$ and choose $a\in A\setminus B$ . Since $p$ satisfies the valuated basis exchange axiom (2) there exists $b\in B\setminus A$ such that

$$\begin{eqnarray}p(A)+p(B)\geqslant p(A\cup b-a)+p(B\cup a-b).\end{eqnarray}$$

Subtracting $\sum _{e\in A}w_{e}+\sum _{e\in B}w_{e}$ on both sides we get

$$\begin{eqnarray}p_{\mathbf{w}}(A)+p_{\mathbf{w}}(B)\geqslant p_{\mathbf{w}}(A\cup b-a)+p_{\mathbf{w}}(B\cup a-b).\end{eqnarray}$$

This implies that $p_{\mathbf{w}}$ also satisfies the valuated basis exchange axiom. Moreover, if $A,B\in \operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})$ , then both $A\cup b-a$ and $B\cup a-b$ are in $\operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})$ as well.

We now prove that the circuits of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ have the desired description. The $\mathbf{w}$ -initial term of any fundamental circuit $H(B,e)$ of ${\mathcal{M}}$ (as in (3)) is the set of $e^{\prime }\in E$ for which $p(B\cup e-e^{\prime })-p(B)+w_{e^{\prime }}$ is minimal. Adding $p(B)-\sum _{a\in B\cup e}w_{a}$ , we get

(4) $$\begin{eqnarray}\operatorname{in}_{\mathbf{w}}H(B,e)=\{e^{\prime }\in E:p_{\boldsymbol{ w}}(B\cup e-e^{\prime })\text{ is minimal among all }e^{\prime }\in E\}.\end{eqnarray}$$

Any circuit $C$ of the initial matroid $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ is the fundamental circuit of an element $e\in E$ over a basis $B\in \operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})$ , and thus has the form

$$\begin{eqnarray}\displaystyle C & = & \displaystyle \{e^{\prime }\in E:B\cup e-e^{\prime }\in \operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})\}\nonumber\\ \displaystyle & = & \displaystyle \{e^{\prime }\in E:p_{\mathbf{w}}(B\cup e-e^{\prime })=p_{\mathbf{w}}(B)\}\nonumber\\ \displaystyle & = & \displaystyle \operatorname{in}_{\mathbf{w}}H(B,e).\nonumber\end{eqnarray}$$

This shows that all circuits of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ appear in the collection $\operatorname{in}_{\mathbf{w}}{\mathcal{C}}({\mathcal{M}})$ . To prove our claim, it then suffices to show that each set in $\operatorname{in}_{\mathbf{w}}{\mathcal{C}}({\mathcal{M}})$ is a dependent set of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ . Assume by contradiction that $\operatorname{in}_{\mathbf{w}}H\subseteq B$ for some $H\in {\mathcal{C}}({\mathcal{M}})$ and some basis $B\in \operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})$ . Fix $e\in \operatorname{in}_{\mathbf{w}}H$ , and a basis $B^{\prime }$ of the underlying matroid $\text{}\underline{{\mathcal{M}}}$ such that $H=\unicode[STIX]{x1D706}\odot H(B^{\prime },e)$ for some $\unicode[STIX]{x1D706}\in \mathbb{R}$ . Since $e\in B\setminus B^{\prime }$ , the valuated basis exchange axiom for $p_{\mathbf{w}}$ implies that there exists $e^{\prime }\in B^{\prime }\setminus B$ such that

(5) $$\begin{eqnarray}p_{\mathbf{w}}(B)+p_{\mathbf{w}}(B^{\prime })\geqslant p_{\boldsymbol{ w}}(B\cup e^{\prime }-e)+p_{\boldsymbol{ w}}(B^{\prime }\cup e-e^{\prime }).\end{eqnarray}$$

As $\operatorname{in}_{\mathbf{w}}H(B^{\prime },e)\subseteq B$ and $e^{\prime }\not \in B$ , by (4) we have $p_{\mathbf{w}}(B^{\prime })<p_{\mathbf{w}}(B^{\prime }\cup e-e^{\prime })$ . Moreover, since $B\in \operatorname{in}_{\mathbf{w}}{\mathcal{B}}({\mathcal{M}})$ , we also have $p_{\mathbf{w}}(B)\leqslant p_{\mathbf{w}}(B\cup e^{\prime }-e)$ . But then, adding these two inequalities gives a contradiction to (5).

We now prove that the set of cycles of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ is equal to $\operatorname{in}_{\mathbf{w}}{\mathcal{V}}({\mathcal{M}})$ . Any $H\in {\mathcal{V}}({\mathcal{M}})$ has the form $H=H^{1}\oplus \cdots \oplus H^{k}$ for circuits $H^{1},\ldots ,H^{k}$ of ${\mathcal{M}}$ . Its initial form $\operatorname{in}_{\mathbf{w}}H$ is the union of those initial forms $\operatorname{in}_{\mathbf{w}}H^{i}$ for which $\min _{e\in E}(H_{e}^{i}+w_{e})=\min _{e\in E}(H_{e}+w_{e})$ . It follows that the collection $\operatorname{in}_{\mathbf{w}}{\mathcal{V}}({\mathcal{M}})$ is closed under unions. Moreover, our previous claim implies that $\operatorname{in}_{\mathbf{w}}{\mathcal{V}}({\mathcal{M}})$ contains all circuits of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ . To complete the proof, it is thus enough to show that, for any circuit $G$ of ${\mathcal{M}}$ , the set $\operatorname{in}_{\mathbf{w}}G$ is a cycle of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ .

We proceed by contradiction. Assume $G$ is a circuit of ${\mathcal{M}}$ such that $\operatorname{in}_{\mathbf{w}}G$ is not a cycle, and take $G$ such that $\operatorname{in}_{\mathbf{w}}G$ is inclusion-minimal with this property. Since $\operatorname{in}_{\mathbf{w}}G$ is dependent in $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ , it contains a circuit $C$ of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ , which must have the form $C=\operatorname{in}_{\mathbf{w}}H$ for some circuit $H$ of ${\mathcal{M}}$ . After tropically rescaling, we may assume that $\min _{e\in E}(G_{e}+w_{e})=\min _{e\in E}(H_{e}+w_{e})$ . Let $e\in \operatorname{in}_{\mathbf{w}}H\subseteq \operatorname{in}_{\mathbf{w}}G$ . We have $G_{e}+w_{e}=H_{e}+w_{e}$ , and so $G_{e}=H_{e}$ . Let $e^{\prime }\in \operatorname{in}_{\mathbf{w}}G$ be such that $e^{\prime }$ is in no circuit of $\operatorname{in}_{\mathbf{w}}{\mathcal{M}}$ contained in $\operatorname{in}_{\mathbf{w}}G$ ; in particular, $e^{\prime }\notin \operatorname{in}_{\mathbf{w}}H$ . It follows that $G_{e^{\prime }}+w_{e^{\prime }}<H_{e^{\prime }}+w_{e^{\prime }}$ , and so $G_{e^{\prime }}<H_{e^{\prime }}$ . We can then apply the circuit elimination axiom to get a circuit $F$ of ${\mathcal{M}}$ such that $F_{e}=\infty$ , $F_{e^{\prime }}=G_{e^{\prime }}$ , and $F\geqslant G\oplus H$ . This implies that $e^{\prime }\in \operatorname{in}_{\mathbf{w}}F$ , $\operatorname{in}_{\mathbf{w}}F\subseteq \operatorname{in}_{\mathbf{w}}G$ , and $e\notin \operatorname{in}_{\mathbf{w}}F$ . We thus have that $\operatorname{in}_{\mathbf{w}}F$ is a proper subset of $\operatorname{in}_{\mathbf{w}}G$ , and our minimality assumption implies that $\operatorname{in}_{\mathbf{w}}F$ is a cycle. This contradicts our choice of $e^{\prime }$ , as $e^{\prime }\in \operatorname{in}_{\mathbf{w}}F$ .◻

In order to use Lemma 3.3 to describe the relationship between the valuated matroids of an ideal and those of its initial ideals, we need the notion of contraction of a valuated matroid.

Let ${\mathcal{M}}$ be a valuated matroid on the ground set $E$ , and let $A$ be a subset of $E$ of rank $s$ . Fix a basis $B_{A}$ of the restriction $\text{}\underline{{\mathcal{M}}}|A$ . The contraction ${\mathcal{M}}/A$ with respect to $B_{A}$ is the valuated matroid ${\mathcal{M}}/A=(E\setminus A,q)$ of rank $r-s$ with basis valuation function $q:\binom{E\setminus A}{r-s}\rightarrow R$ given by

$$\begin{eqnarray}q(B)=p(B\sqcup B_{A}).\end{eqnarray}$$

If we replace $B_{A}$ by a different basis $B_{A}^{\prime }$ of $A$ , then $q$ changes by global (tropical) multiplication by a nonzero constant; see, for instance, [Reference MurotaMur10, Theorem 5.2.5]. The resulting valuated matroids are thus the same, and we suppress the choice of $B_{A}$ from the notation for contractions. The set of vectors of the contraction ${\mathcal{M}}/A$ is

$$\begin{eqnarray}{\mathcal{V}}({\mathcal{M}}/A)=\{H[E\setminus A]:H\in {\mathcal{V}}({\mathcal{M}})\},\end{eqnarray}$$

where $H[E\setminus A]\in R^{E\setminus A}$ is given by $(H[E\setminus A])_{e}:=H_{e}$ for any $e\in E\setminus A$ ; see, for example, [Reference BakerBak16, Theorem 4.17(2)].

If $I$ is a homogeneous tropical ideal in $\mathbb{B}[x_{0},\ldots ,x_{n}]$ , we write $M_{d}(I):=\text{}\underline{{\mathcal{M}}_{d}(I)}$ to emphasize the fact that no extra information is encoded in the valuated matroid ${\mathcal{M}}_{d}(I)$ besides its underlying matroid. Also, if $M$ is a matroid on the ground set $E$ , and $F$ is a finite set, the coloop extension $M\oplus F$ is the matroid on the ground set $E\sqcup F$ obtained by attaching to $M$ all the elements of $F$ as coloops, i.e., ${\mathcal{B}}(M\oplus F)=\{B\sqcup F:B\in {\mathcal{B}}(M)\}$ .

We now apply these results to study Hilbert functions of homogeneous tropical ideals in $R[x_{0},\ldots ,x_{n}]$ , where $R$ is again a general valuative semifield, and to prove that they satisfy the ascending chain condition.

Definition 3.5 (Hilbert functions).

Let $I$ be a homogeneous tropical ideal in $R[x_{0},\ldots ,x_{n}]$ . Its Hilbert function is the function $H_{I}:\mathbb{Z}_{{\geqslant}0}\rightarrow \mathbb{Z}_{{\geqslant}0}$ defined by

$$\begin{eqnarray}H_{I}(d):=\operatorname{rank}({\mathcal{M}}_{d}(I)).\end{eqnarray}$$

If $J\subseteq K[x_{0},\ldots ,x_{n}]$ is a homogeneous ideal, then the matroid $\text{}\underline{{\mathcal{M}}_{d}(\operatorname{trop}(J))}$ encodes the dependencies in $(K[x_{0},\ldots ,x_{n}]/J)_{d}$ among the monomials in $\operatorname{Mon}_{d}$ ; a subset of $\operatorname{Mon}_{d}$ is dependent if and only if there is a polynomial $f\in J_{d}$ whose support is contained in this subset. This implies that the bases of $\text{}\underline{{\mathcal{M}}_{d}(\operatorname{trop}(J))}$ correspond to monomial bases of the vector space $(K[x_{0},\ldots ,x_{n}]/J)_{d}$ . It follows that Hilbert functions are preserved under tropicalization:

$$\begin{eqnarray}H_{\operatorname{trop}(J)}(d)=\dim _{K}((K[x_{0},\ldots ,x_{n}]/J)_{d})\end{eqnarray}$$

for any $d\in \mathbb{Z}_{{\geqslant}0}$ . See also [Reference Giansiracusa and GiansiracusaGG16, Theorem 7.1.6].

A key fact about standard Gröbner theory is that the Hilbert function of an ideal is preserved when passing to an initial ideal. We next observe that this also holds in the tropical setting. Note that $\mathbf{w}$ is not allowed to have infinite coordinates for this result.

The following result will be useful in our study of tropical ideals and their Hilbert functions.

In § 5 we will see that, in fact, the set of $\mathbf{w}\in \mathbb{R}^{n+1}$ for which $\operatorname{in}_{\mathbf{w}}(I)$ is not generated by monomials is a finite union of polyhedra of codimension at least 1.

The following result shows that Hilbert functions of tropical ideals are eventually polynomial.

The following example shows that tropical ideals are typically not finitely generated. However, we prove in Theorem 3.11 that they do have some Noetherian properties, in the sense that they satisfy the ascending chain condition.

Example 3.10 (Tropical ideals need not be finitely generated).

Let $J:=\langle x-y\rangle \subseteq K[x,y]$ , and let $I:=\operatorname{trop}(J)\subseteq R[x,y]$ . We claim that the homogeneous tropical ideal $I$ is not finitely generated. Note that $x^{d}-y^{d}\in J$ for all $d\geqslant 1$ , so $x^{d}\oplus y^{d}\in I$ for all $d\geqslant 1$ . Suppose that $f_{1},\ldots ,f_{r}$ is a finite homogeneous generating set for $I$ . Then for all $d\geqslant 1$ we can write

(6) $$\begin{eqnarray}x^{d}\oplus y^{d}=\bigoplus h_{id}\odot f_{i}.\end{eqnarray}$$

Since there is no cancelation in $R[x,y]$ , if $m$ is a monomial occurring in $h_{id}$ for some $d$ , and $m^{\prime }$ is a monomial occurring in $f_{i}$ , then $mm^{\prime }$ occurs in $h_{id}\odot f_{i}$ , and thus in $x^{d}\oplus y^{d}$ . This is only possible if each $f_{i}$ is either $x^{d}\oplus y^{d}$ or a power of $x$ or $y$ . Thus for any $d\geqslant \max (\deg (f_{i}))$ , each $f_{i}$ occurring in (6) must be a power of $x$ or $y$ , and at least one of each must occur. This means that $I_{d}=R[x,y]_{d}$ for $d\gg 0$ , and so the Hilbert function of $I$ equals zero for $d\gg 0$ , which contradicts the fact that the Hilbert functions of $I$ and $J$ agree.

4 Subschemes of tropical toric varieties

We now extend the definition of tropical ideals to cover subschemes of tropical toric varieties other than tropical projective space.

Tropical toric varieties are defined analogously to classical toric varieties. Given a rational polyhedral fan $\unicode[STIX]{x1D6F4}$ , we associate an affine tropical toric variety to each cone, and glue these together. Explicitly, fix a lattice $N\cong \mathbb{Z}^{n}$ with dual lattice $M:=\operatorname{Hom}(N,\mathbb{Z})\cong \mathbb{Z}^{n}$ . Given a rational polyhedral fan $\unicode[STIX]{x1D6F4}$ in $N_{\mathbb{R}}:=N\otimes \mathbb{R}\cong \mathbb{R}^{n}$ , to each $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ we associate the vector space $N(\unicode[STIX]{x1D70E}):=N_{\mathbb{R}}/\operatorname{span}(\unicode[STIX]{x1D70E})$ . The tropical toric variety associated to $\unicode[STIX]{x1D6F4}$ is

$$\begin{eqnarray}\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}):=\coprod _{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}}N(\unicode[STIX]{x1D70E}).\end{eqnarray}$$

To each $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ we also associate the space $U_{\unicode[STIX]{x1D70E}}^{\operatorname{trop}}:=\operatorname{Hom}(\unicode[STIX]{x1D70E}^{\vee }\,\cap \,M,\overline{\mathbb{R}})$ of semigroup homomorphisms to the semigroup $(\overline{\mathbb{R}},\odot )$ . We have $U_{\unicode[STIX]{x1D70E}}^{\operatorname{trop}}=\coprod _{\unicode[STIX]{x1D70F}\preccurlyeq \unicode[STIX]{x1D70E}}N(\unicode[STIX]{x1D70F})$ , where the union is over all faces $\unicode[STIX]{x1D70F}$ of $\unicode[STIX]{x1D70E}$ . The topology on $\overline{\mathbb{R}}$ has as basic open sets all open intervals in $\mathbb{R}$ , plus intervals of the form $(a,\infty ]=\{b\in \mathbb{R}:b>a\}\cup \{\infty \}$ . This induces the product topology on $\overline{\mathbb{R}}^{\unicode[STIX]{x1D70E}^{\vee }\cap M}$ , and thus on $U_{\unicode[STIX]{x1D70E}}^{\operatorname{trop}}$ . See [Reference KajiwaraKaj08, Reference PaynePay09, Reference RabinoffRab12] or [Reference Maclagan and SturmfelsMS15, ch. 6] for more details on this construction. Important special cases are the tropical versions of affine space, projective space, and the torus; see Example 4.2.

One can also tropicalize the notion of Cox ring of a toric variety. In the case that the fan $\unicode[STIX]{x1D6F4}$ does not span all of $N_{\mathbb{R}}$ , for ease of notation we tropicalize the version given in [Reference Cox, Little and SchenckCLS11, (5.1.10)]. The more invariant choice given in [Reference Cox, Little and SchenckCLS11, Theorem 5.1.17] can also be tropicalized.

In what follows we will only consider rational polyhedral fans, so will refer to them simply as fans.

Definition 4.1 (Cox semirings of tropical toric varieties).

Let $\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})$ be a tropical toric variety defined by a fan $\unicode[STIX]{x1D6F4}\subseteq N_{\mathbb{R}}\cong \mathbb{R}^{n}$ . Let $s$ be the number of rays of $\unicode[STIX]{x1D6F4}$ , and let $t=n-\dim (\operatorname{span}(\unicode[STIX]{x1D6F4}))$ . Write $m=s+t$ . The Cox semiring of $\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})$ is the semiring

$$\begin{eqnarray}\operatorname{Cox}(\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})):=\overline{\mathbb{R}}[x_{1},\ldots ,x_{m}].\end{eqnarray}$$

Write $N$ as the direct sum $N^{\prime }\oplus N^{\prime \prime }$ , where the rays of $\unicode[STIX]{x1D6F4}$ span $N_{\mathbb{R}}^{\prime }$ . Fix an $m\times n$ matrix $Q$ whose $i$ th row is the first lattice point $\mathbf{v}_{i}$ on the $i$ th ray of $\unicode[STIX]{x1D6F4}$ for $1\leqslant i\leqslant s$ , and whose last $t$ rows form a basis for $N^{\prime \prime }$ . We grade $\operatorname{Cox}(\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}))$ by the ‘combinatorial Chow group’ $A_{n-1}(\unicode[STIX]{x1D6F4}):=\operatorname{coker}(M\cong \mathbb{Z}^{n}\stackrel{\unicode[STIX]{x1D711}}{\rightarrow }\mathbb{Z}^{m}),$ where the map $\unicode[STIX]{x1D711}$ is given by $\unicode[STIX]{x1D711}(\mathbf{u})=Q\mathbf{u}$ . This gives the last $t$ variables of $\operatorname{Cox}(\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}))$ degree $\mathbf{0}\in A_{n-1}(\unicode[STIX]{x1D6F4})$ .

We will identify the rays of $\unicode[STIX]{x1D6F4}$ with the elements of $\{1,\ldots ,s\}$ , and thus general cones of $\unicode[STIX]{x1D6F4}$ with subsets of $\{1,\ldots ,s\}$ . For a cone $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ write

$$\begin{eqnarray}\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}=\mathop{\prod }_{i\not \in \unicode[STIX]{x1D70E}}x_{i}\in \overline{\mathbb{R}}[x_{1},\ldots ,x_{m}].\end{eqnarray}$$

The localization of the semiring $\overline{\mathbb{R}}[x_{1},\ldots ,x_{m}]$ at a monomial is defined analogously to the ring case. The degree-0 part of the localization of the Cox semiring at $\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}$ is isomorphic to the semigroup semiring

$$\begin{eqnarray}(\operatorname{Cox}(\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}))_{\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}})_{\mathbf{0}}\cong \overline{\mathbb{R}}[\unicode[STIX]{x1D70E}^{\vee }\cap M].\end{eqnarray}$$

Note that the last $t$ variables are always inverted in these localizations.

Classical toric varieties can alternatively be described by the quotient construction $X_{\unicode[STIX]{x1D6F4}}=(\mathbb{A}^{m}\setminus V(B_{\unicode[STIX]{x1D6F4}}))/H_{\unicode[STIX]{x1D6F4}}$ , where $B_{\unicode[STIX]{x1D6F4}}$ denotes the irrelevant ideal $\langle \mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}:\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}\rangle \subseteq K[x_{1},\ldots ,x_{m}]$ , and $H_{\unicode[STIX]{x1D6F4}}=\operatorname{Hom}(A_{n-1}(X_{\unicode[STIX]{x1D6F4}}),K^{\ast })$ . This construction also tropicalizes [Reference Maclagan and SturmfelsMS15, Proposition 6.2.6]:

(7) $$\begin{eqnarray}\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})=(\overline{\mathbb{R}}^{m}\setminus V(\operatorname{trop}(B_{\unicode[STIX]{x1D6F4}})))/\operatorname{trop}(H_{\unicode[STIX]{x1D6F4}}),\end{eqnarray}$$

where $\operatorname{trop}(B_{\unicode[STIX]{x1D6F4}})$ denotes the monomial ideal in $\overline{\mathbb{R}}[x_{1},\ldots ,x_{m}]$ given by

$$\begin{eqnarray}\operatorname{trop}(B_{\unicode[STIX]{x1D6F4}})=\langle \mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}:\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}\rangle\end{eqnarray}$$

and $\operatorname{trop}(H_{\unicode[STIX]{x1D6F4}})$ is the $n$ -dimensional subspace $\ker (Q^{T})\subseteq \mathbb{R}^{m}$ . Write

$$\begin{eqnarray}\overline{\mathbb{R}}_{\unicode[STIX]{x1D70E}}^{m}:=\{\mathbf{w}\in \overline{\mathbb{R}}^{m}:w_{i}=\infty \text{for }i\in \unicode[STIX]{x1D70E},~w_{i}\neq \infty \text{for }i\notin \unicode[STIX]{x1D70E}\}.\end{eqnarray}$$

The equality in (7) identifies $U_{\unicode[STIX]{x1D70E}}^{\operatorname{trop}}$ with the subset $(\overline{\mathbb{R}}^{m}\setminus V(\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}))/\ker (Q^{T})$ , and $N(\unicode[STIX]{x1D70E})$ with $(\overline{\mathbb{R}}_{\unicode[STIX]{x1D70E}}^{m}\setminus V(\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}))/\ker (Q^{T})$ . The topology on $\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})$ is consistent with the quotient topology coming from the topology on $\overline{\mathbb{R}}^{m}$ .

Example 4.2 (Tropical affine space, projective space, and the torus).

The fan $\unicode[STIX]{x1D6F4}$ consisting of the positive orthant in $N_{\mathbb{R}}\cong \mathbb{R}^{n}$ gives rise to the toric variety $\mathbb{A}^{n}$ . The tropical toric variety $\operatorname{trop}(\mathbb{A}^{n})$ is $\overline{\mathbb{R}}^{n}$ . Any face of the positive orthant has the form $\operatorname{cone}\{\mathbf{e}_{i}:i\in \unicode[STIX]{x1D70E}\}$ for $\unicode[STIX]{x1D70E}\subseteq \{1,\ldots ,n\}$ ; the stratum $N(\unicode[STIX]{x1D70E})$ corresponds precisely to the set $\overline{\mathbb{R}}_{\unicode[STIX]{x1D70E}}^{n}\subseteq \overline{\mathbb{R}}^{n}$ . The Cox semiring $\operatorname{Cox}(\operatorname{trop}(\mathbb{A}^{n}))$ is equal to $\overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ with the trivial grading $\deg (x_{i})=0$ for all $i$ .

The fan $\unicode[STIX]{x1D6F4}$ defining $\mathbb{P}^{n}$ has rays spanned by $\mathbf{e}_{1},\ldots ,\mathbf{e}_{n}$ and $\mathbf{e}_{0}=-\sum _{i=1}^{n}\mathbf{e}_{i}$ in $N_{\mathbb{R}}\cong \mathbb{R}^{n}$ , and cones spanned by any $j$ of these rays for $j\leqslant n$ . The Cox semiring in this case is $\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]$ with the standard grading $\deg (x_{i})=1$ for all $i$ . The tropicalization $\operatorname{trop}(\mathbb{P}^{n})$ equals $(\overline{\mathbb{R}}^{n+1}\setminus \{(\infty ,\ldots ,\infty )\})/\mathbb{R}\mathbf{1}$ , where $\mathbf{1}=(1,\ldots ,1)$ . Figure 2 shows the fan $\unicode[STIX]{x1D6F4}$ and its corresponding tropical toric variety $\operatorname{trop}(\mathbb{P}^{2})=\coprod _{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}}N(\unicode[STIX]{x1D70E})$ in the case where $n=2$ .

Figure 2. Tropical projective space $\operatorname{trop}(\mathbb{P}^{2})$ as a tropical toric variety.

The fan $\unicode[STIX]{x1D6F4}$ consisting of just the origin in $N_{\mathbb{R}}\cong \mathbb{R}^{n}$ gives rise to the tropical torus $\mathbb{R}^{n}$ . Its Cox semiring is $\overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ with the trivial grading $\deg (x_{i})=0$ for all $i$ . The localization at the unique cone $\unicode[STIX]{x1D70E}=\{\mathbf{0}\}$ is $\overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]_{x_{1}\cdots x_{n}}\cong \overline{\mathbb{R}}[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}]$ .

Subschemes of a classical toric variety can be described by ideals in its Cox ring. We now extend this to the tropical realm.

For a tropical polynomial $f=\sum a_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}\in \overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ , the homogenization of $f$ is the polynomial $\tilde{f}:=\sum a_{\mathbf{u}}\odot x_{0}^{d-|\mathbf{u}|}\odot \mathbf{x}^{\mathbf{u}}\in \overline{\mathbb{R}}[x_{0},x_{1},\ldots ,x_{n}]$ , where $d=\max _{a_{\mathbf{u}}\neq \infty }|\mathbf{u}|$ . The homogenization of an ideal $I\subseteq \overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ is $\tilde{I} :=\langle \tilde{f}:f\in I\rangle$ . This has the property that every homogeneous polynomial $g\in \tilde{I}$ has the form $x_{0}^{m}\odot \tilde{f}$ for some $m\geqslant 0$ and $f\in I$ , and so the set of homogeneous polynomials of degree $d$ in $\tilde{I}$ is in bijection with the set of polynomials in $I$ of degree at most $d$ . This implies that, if $I\subseteq \overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ is a (locally) tropical ideal, then $\tilde{I}$ is a homogeneous tropical ideal.

The following example shows that the converse of Proposition 4.4 is not true.

When working with semirings, congruences perform some of the functions of ideals in rings. A congruence on a semiring $S$ is an equivalence relation ${\sim}$ on $S$ that is compatible with addition and multiplication: $f\sim g$ and $f^{\prime }\sim g^{\prime }$ imply that $(f\oplus f^{\prime })\sim (g\oplus g^{\prime })$ and $(f\odot g)\sim (g\odot g^{\prime })$ . In [Reference Giansiracusa and GiansiracusaGG16], Jeffrey and Noah Giansiracusa defined a congruence associated to an ideal in the tropical setting, as we now recall.

Definition 4.10 (Bend relations [Reference Giansiracusa and GiansiracusaGG16]).

Let $S:=\overline{\mathbb{R}}[\unicode[STIX]{x1D70E}^{\vee }\cap M]$ for a cone $\unicode[STIX]{x1D70E}\subseteq N_{\mathbb{R}}$ . For a polynomial $f=\bigoplus a_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}\in S$ , denote by $f_{\hat{\mathbf{v}}}$ the polynomial obtained by deleting the term involving $\mathbf{x}^{\mathbf{v}}$ from $f$ , i.e., $f_{\hat{\mathbf{v}}}=\bigoplus _{\mathbf{u}\neq \mathbf{v}}a_{\mathbf{u}}\odot \mathbf{x}^{\mathbf{u}}$ . The bend relations of $f$ are

$$\begin{eqnarray}B(f):=\{f\sim f_{\hat{\mathbf{v}}}:\mathbf{x}^{\mathbf{v}}\in \operatorname{supp}(f)\}.\end{eqnarray}$$

The bend congruence ${\mathcal{B}}(I)$ of an ideal $I\subseteq S$ is the congruence on $S$ ,

$$\begin{eqnarray}{\mathcal{B}}(I):=\langle B(f):f\in I\rangle .\end{eqnarray}$$

The coordinate semiring associated to $I$ is $S/{\mathcal{B}}(I)$ .

This construction is motivated by the fact that, for any ideal $I\subseteq \overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ , there is a canonical isomorphism $\operatorname{Hom}(S/{\mathcal{B}}(I),\overline{\mathbb{R}})\cong V(I)\,\subseteq \overline{\mathbb{R}}^{n}$ .

In [Reference Giansiracusa and GiansiracusaGG16], the tropicalization of the subscheme of $\mathbb{A}^{n}$ defined by an ideal $J\subseteq K[x_{1},\ldots ,x_{n}]$ is defined to be $\operatorname{Spec}(\overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]/{\mathcal{B}}(\operatorname{trop}(J)))$ . Here $\operatorname{Spec}$ is used in the sense of $\mathbb{F}_{1}$ -geometry, so it denotes a topological space with a sheaf of semirings. The underlying space of $\operatorname{Spec}(R)$ for a semiring $R$ is the set of prime ideals of $R$ , where an ideal $I$ is prime if $f\odot g\in I$ implies that $f\in I$ or $g\in I$ . Similarly, the tropicalization of the subscheme of $\mathbb{P}^{n}$ defined by a homogeneous ideal $J\subseteq K[x_{0},\ldots ,x_{n}]$ is $\operatorname{Proj}(\overline{\mathbb{R}}[x_{0},\ldots ,x_{n}]/{\mathcal{B}}(\operatorname{trop}(J)))$ , where the underlying set of $\operatorname{Proj}$ is the set of homogeneous prime ideals not containing $\langle x_{0},\ldots ,x_{n}\rangle$ . This is explained in [Reference LorscheidLor17, §9], where these are presented as a special case of Lorscheid’s blueprints. See also [Reference LorscheidLor12, Reference LorscheidLor15, Reference DurovDur07, Reference Toën and VaquiéTV09] for related work. The results in [Reference Giansiracusa and GiansiracusaGG16] are set in this context, so they describe the tropicalization of any subscheme of a classical toric variety. We now generalize this construction to include more general subschemes of tropical toric varieties that are not necessarily tropicalizations.

Definition 4.11 (Subschemes of tropical toric varieties).

Let $I$ be a locally tropical ideal of the Cox semiring $S$ of a tropical toric variety $\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})$ . The subscheme of $\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})$ defined by $I$ is given locally as a subscheme of $U_{\unicode[STIX]{x1D70E}}^{\operatorname{trop}}$ by

$$\begin{eqnarray}\operatorname{Spec}(\overline{\mathbb{R}}[\unicode[STIX]{x1D70E}^{\vee }\cap M]/{\mathcal{B}}((IS_{\boldsymbol{ x}^{\hat{\unicode[STIX]{x1D70E}}}})_{\mathbf{0}})).\end{eqnarray}$$

Definition 4.11 makes sense without the restriction that $I$ be a locally tropical ideal; the reason for requiring this condition is that, as shown in Example 5.14, without it the variety of the subscheme might not be a finite polyhedral complex.

Example 4.13 (Tropical prime ideals of $\overline{\mathbb{R}}[x]$ ).

We classify all tropical prime ideals in the semiring $\overline{\mathbb{R}}[x]$ of tropical polynomials in one variable. For any $a\in \overline{\mathbb{R}}$ , let $J_{a}$ be the tropical ideal consisting of all tropical polynomials $f$ whose variety $V(f)\subseteq \overline{\mathbb{R}}$ contains $a$ (see Example 4.6). If $a=\infty$ , then $J_{a}$ is the ideal of polynomials with no constant term. The fact that any two tropical polynomials $f$ and $g$ satisfy $V(f\odot g)=V(f)\cup V(g)$ implies that $J_{a}$ is a tropical prime ideal. We will prove that, in fact, these are the only proper nontrivial prime tropical ideals in $\overline{\mathbb{R}}[x]$ .

Fix a prime tropical ideal $P\subseteq \overline{\mathbb{R}}[x]$ , and let $f=\bigoplus _{i=0}^{d}b_{i}\odot x^{i}\in P$ , where $b_{d}\neq \infty$ . Let $V(f)=\{a_{1},\ldots ,a_{s}\}$ . The multiplicity $m_{i}$ of $a_{i}$ is the maximum of $j_{2}-j_{1}$ where both $j_{1}$ and $j_{2}$ achieve the minimum in $\min _{j}(b_{j}+ja_{i})$ . Set $g:=b_{d}\odot \prod _{i=1}^{s}(x\oplus a_{i})^{m_{i}}$ . The polynomial $g$ is the tropical polynomial with smallest coefficients for which $f(z)=g(z)$ for all $z\in \overline{\mathbb{R}}$ ; see, for example, [Reference Grigg and ManwaringGM07, §4]. Writing $g=\bigoplus _{j=0}^{d}c_{j}\odot x^{j}$ , we have

$$\begin{eqnarray}c_{j}=\min \{b_{j}\}\cup \biggl\{\frac{b_{i}\cdot (k-j)+b_{k}\cdot (j-i)}{k-i}:0\leqslant i<j<k\leqslant d\biggr\},\end{eqnarray}$$

for any $j$ , as stated in [Reference Grigg and ManwaringGM07, Lemma 3.3]. In particular, $c_{0}=b_{0}$ and $c_{d}=b_{d}$ . The two consequences of this formula that we will use are that $c_{i}\leqslant b_{i}$ for all $i$ , and, if $c_{i}<b_{i}$ , then there is $l$ with $c_{i-1}-a_{l}=c_{i}=c_{i+1}+a_{l}$ .

We claim that $g\odot f=g^{2}$ , and so $g^{2}\in P$ . To show this we need to prove that for all $0\leqslant k\leqslant 2d$ we have $\min _{i+j=k}(b_{i}+c_{j})=\min _{i+j=k}(c_{i}+c_{j})$ . The inequality ${\geqslant}$ follows from the fact that $b_{i}\geqslant c_{i}$ for all $i$ . Suppose now that there is a pair $(i,j)$ with $c_{i}+c_{j}=\min _{i^{\prime }+j^{\prime }=i+j}(c_{i^{\prime }}+c_{j^{\prime }})<\min _{i^{\prime }+j^{\prime }=i+j}(b_{i^{\prime }}+c_{j^{\prime }})$ . This implies that $c_{i}<b_{i}$ and $c_{j}<b_{j}$ , so $0<i,j<d$ and there must be $l$ and $l^{\prime }$ with $c_{i-1}-a_{l}=c_{i}=c_{i+1}+a_{l}$ and $c_{j-1}-a_{l^{\prime }}=c_{j}=c_{j+1}+a_{l^{\prime }}$ . Without loss of generality we may assume that $a_{l}\leqslant a_{l^{\prime }}$ . We have $c_{i}+c_{j}=c_{i-1}-a_{l}+c_{j+1}+a_{l^{\prime }}$ , so $c_{i-1}+c_{j+1}\leqslant c_{i}+c_{j}$ . We may thus replace $(i,j)$ by $(i-1,j+1)$ and repeat. After a finite number of iterations we will have a pair $(i,j)$ with $c_{i}+c_{j}$ achieving the minimum $\min _{i^{\prime }+j^{\prime }=i+j}(c_{i^{\prime }}+c_{j^{\prime }})$ and at least one of $c_{i}=b_{i}$ or $c_{j}=b_{j}$ , which is a contradiction. This proves that $g^{2}=g\odot f$ .

Since $P$ is prime, the fact that $g^{2}\in P$ implies that $x\oplus a\in P$ for some $a=a_{i}$ . Multiplying by powers of $x$ we see that $P$ must contain all polynomials of the form $x^{j}\oplus a\odot x^{j-1}$ . Furthermore, the vector elimination axiom forces $P$ to contain all polynomials of the form $a^{m}\odot x^{l}\oplus a^{l}\odot x^{m}$ . Since these polynomials generate the ideal $J_{a}$ , we see that $J_{a}\subseteq P$ . If $P$ were strictly larger than $J_{a}$ , it would contain a polynomial $f^{\prime }$ with $a\not \in V(f^{\prime })$ . The argument above then shows that $x\oplus b\in P$ for some $b\in V(f^{\prime })$ . But then the vector elimination axiom applied to $x\oplus a$ and $x\oplus b$ forces $P$ to contain the constant $\min (a,b)$ , which implies that $P$ is the unit ideal.

We conclude this section by extending the notion of variety to ideals in a Cox semiring $\operatorname{Cox}(\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}))$ . For a fan $\unicode[STIX]{x1D6F4}\subseteq N_{\mathbb{R}}\cong \mathbb{R}^{n}$ and $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}$ , the coordinate ring of the corresponding stratum $N(\unicode[STIX]{x1D70E})$ is

$$\begin{eqnarray}(S_{\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}}/\langle x_{i}\sim \infty :i\in \unicode[STIX]{x1D70E}\rangle )_{\mathbf{0}}\cong \overline{\mathbb{R}}[\unicode[STIX]{x1D70E}^{\bot }\cap M]\cong \overline{\mathbb{R}}[y_{1}^{\pm 1},\ldots ,y_{n-\dim (\unicode[STIX]{x1D70E})}^{\pm 1}].\end{eqnarray}$$

The variety of an ideal $I\subseteq \overline{\mathbb{R}}[y_{1}^{\pm 1},\ldots ,y_{l}^{\pm 1}]$ is

$$\begin{eqnarray}V(I):=\{\mathbf{w}\in \mathbb{R}^{l}:\text{the min in }f(\mathbf{w})\text{ is achieved at least twice for all }f\in I\setminus \{\infty \}\}.\end{eqnarray}$$

For $f\in \overline{\mathbb{R}}[y_{1}^{\pm 1},\ldots ,y_{l}^{\pm 1}]$ the initial term $\operatorname{in}_{\mathbf{w}}(f)$ is defined as in Definition 3.1. Similarly, for an ideal $I\subseteq \overline{\mathbb{R}}[y_{1}^{\pm 1},\ldots ,y_{l}^{\pm 1}]$ , its initial ideal is $\operatorname{in}_{\mathbf{w}}(I):=\langle \operatorname{in}_{\mathbf{w}}(f):f\in I\rangle \subseteq \mathbb{B}[y_{1}^{\pm 1},\ldots ,y_{l}^{\pm 1}]$ .

Definition 4.14 (Varieties of ideals in Cox semirings).

Fix a fan $\unicode[STIX]{x1D6F4}\subseteq N_{\mathbb{R}}\cong \mathbb{R}^{n}$ , and let $I$ be a homogeneous ideal in $S:=\operatorname{Cox}(\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}))$ . We define

$$\begin{eqnarray}V_{\unicode[STIX]{x1D70E}}(I):=V((IS_{\mathbf{x}^{\hat{\unicode[STIX]{x1D70E}}}}/\langle x_{i}\sim \infty :i\in \unicode[STIX]{x1D70E}\rangle )_{\mathbf{0}})\subseteq N(\unicode[STIX]{x1D70E})\cong \mathbb{R}^{n-\dim (\unicode[STIX]{x1D70E})}\end{eqnarray}$$

and

$$\begin{eqnarray}V_{\unicode[STIX]{x1D6F4}}(I):=\coprod _{\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6F4}}V_{\unicode[STIX]{x1D70E}}(I)\subseteq \coprod N(\unicode[STIX]{x1D70E})=\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}}).\end{eqnarray}$$

When $\unicode[STIX]{x1D6F4}$ is the positive orthant in $\mathbb{R}^{n}$ , the tropical toric variety $\operatorname{trop}(X_{\unicode[STIX]{x1D6F4}})$ is $\operatorname{trop}(\mathbb{A}^{n})\cong \overline{\mathbb{R}}^{n}$ , and the Cox semiring $S$ equals $\overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]$ with the trivial grading $\deg (x_{i})=0$ for all $i$ . In the next lemma we check that the new notion of variety agrees with that in (1), and also that the easy equivalences of the Fundamental Theorem [Reference Maclagan and SturmfelsMS15, Theorem 3.2.3] hold in this new setting.

Proposition 4.15. Let $\unicode[STIX]{x1D6F4}$ be the positive orthant in $N_{\mathbb{R}}$ , and let $I$ be an ideal in $\overline{\mathbb{R}}[x_{1},\ldots ,x_{n}]=\operatorname{Cox}(\operatorname{trop}(\mathbb{A}^{n}))$ . Then

$$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D6F4}}(I)=V(I) & = & \displaystyle \{\!\mathbf{w}\in \overline{\mathbb{R}}^{n}\mid \text{for all }f\in I\text{ with }f(\mathbf{w})<\infty ,\text{ the minimum in }f(\mathbf{w})\nonumber\\ \displaystyle & & \displaystyle \quad \text{is achieved at least twice}\!\}\!.\nonumber\end{eqnarray}$$

Moreover, we have $\mathbf{w}\in V(I)$ if and only if $\operatorname{in}_{\mathbf{w}}(I)$ does not contain a monomial. Similarly, if $I$ is an ideal in $\overline{\mathbb{R}}[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}]$ and $\mathbf{w}\in \mathbb{R}^{n}$ , then $\mathbf{w}\in V(I)$ if and only if $\operatorname{in}_{\mathbf{w}}(I)\neq \langle 0\rangle$ .