2.1 Homomorphisms and enriched valuations

We will now recall the definition of a hyperfield homomorphism, connecting this with tropical extensions and valuations, and construct several key examples.

Definition 2.14. A map $f\colon\mathbb{H}_1\rightarrow\mathbb{H}_2$ is a hyperfield homomorphism if it satisfies

\begin{align*} f(a \boxplus_1 b) &\subseteq f(a) \boxplus_2 f(b) \, , & f(a \odot_1 b) &= f(a) \odot_2 f(b) \, , \\ f(\mathbb{0}_1) &= \mathbb{0}_2 \, , & f(\mathbb{1}_1) &= \mathbb{1}_2 \, . \end{align*}

An isomorphism of hyperfields is a bijective homomorphism whose inverse map is also a homomorphism.

As with field homomorphisms, it is also straightforward to show that $\mathbb{0}_1$ is the unique element that gets sent to $\mathbb{0}_2$ by f. We can also similarly deduce that $f(a)^{-1} = f(a^{-1})$ and $-f(a) = f(-a)$ for all $a \in \mathbb{H}_1$.

Example 2.16. Every hyperfield has a trivial homomorphism to the Krasner hyperfield, given by

\begin{align*} \omega:\mathbb{H} &\rightarrow \mathbb{K} , \quad \omega(a) = \begin{cases} \mathbb{1}, \quad \text{if} \quad a\neq \mathbb{0}_{\mathbb{H}}. \\ \mathbb{0}, \quad \text{if} \quad a=\mathbb{0}_{\mathbb{H}}. \end{cases} \end{align*}

Example 2.17. Given a factor hyperfield $\mathbb{H}=\mathbb{F}/U$, there is a natural hyperfield homomorphism

\begin{equation*} \tau:\mathbb{F} \rightarrow \mathbb{H} , \quad a \mapsto \mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu \, . \end{equation*}

Given that all the hyperfields we have seen are factor hyperfields, this gives many examples of hyperfield homomorphisms, e.g.

\begin{align*} \operatorname{sgn}:\mathbb{R} &\rightarrow \mathbb{S} \cong \mathbb{R}/\mathbb{R}_{ \gt 0} \, , & \operatorname{ph}:\mathbb{C} &\rightarrow \mathbb{P} \cong \mathbb{C}/\mathbb{R}_{ \gt 0} \, , \\ a &\mapsto \begin{cases} \mathbb{1} & \text{if} \quad a \in \mathbb{R}_{ \gt 0} \\ -\mathbb{1} & \text{if} \quad a \in \mathbb{R}_{ \lt 0} \\ \mathbb{0} & \text{if} \quad a=0 \end{cases} & z &\mapsto \begin{cases} \frac{z}{|z|} & \text{if} \quad z \in \mathbb{C}\backslash\{0\} \\ 0 &\text{if} \quad z=0 \end{cases} \end{align*}

As with tropical extensions of hyperfields, one can consider tropical extensions of homomorphisms.

Definition 2.18. Let $f\colon \mathbb{H}_1 \rightarrow \mathbb{H}_2$ be a map between two hyperfields. Given an ordered abelian group Γ, the tropical extension of f by Γ is the map

\begin{align*} f^\Gamma\colon \mathbb{H}_1\rtimes \Gamma &\rightarrow \mathbb{H}_2\rtimes \Gamma \\ (c,g) &\mapsto (f(c),g) \, . \end{align*}

It is easy to see that if f is a homomorphism, then so is $f^\Gamma$.

Example 2.19. Consider the homomorphism from the tropical complex hyperfield $\mathbb{T}\mathbb{C}$ to the tropical hyperfield:

\begin{align*} \eta \colon \mathbb{T}\mathbb{C} &\rightarrow \mathbb{T} \\ z &\mapsto -\log(|z|) \, . \end{align*}

This is the key example studied in [Reference Maxwell45], as it ‘preserves’ roots of polynomials; we shall discuss this in detail in §3. We can give an alternative description of this map in the language of tropical extensions. As both $\mathbb{T}\mathbb{C} \cong \Phi\rtimes \mathbb{R}$ and $\mathbb{T} \cong \mathbb{K}\rtimes \mathbb{R}$ are tropical extensions, the map can be rephrased as

\begin{align*} \omega^\Gamma \colon \Phi\rtimes \mathbb{R} &\rightarrow \mathbb{K}\rtimes \mathbb{R} \\ (\theta, g) &\mapsto (\mathbb{1},g) \, , \end{align*}

where $\theta = \operatorname{ph}(z)$ and $g = -\log(|z|)$ given by the isomorphism in example 2.9. In particular, η is the extension of the trivial homomorphism $\omega\colon \Phi \rightarrow \mathbb{K}$ by $\mathbb{R}$.

A crucial example of a hyperfield homomorphism will be those coming from valued fields. As this motivates many of the applications of our theory, we recall the definition.

Note that we can demand that valuations are surjective by restricting the range of non-surjective valuations to the value group. This will be beneficial later, allowing us to sidestep certain topological concerns we would have to deal with otherwise.

Example 2.21. A natural example of a valued field is the field of Puiseux series over a field $\mathbb{F}$. These are formal power series whose exponents are rational with a common denominator

\begin{equation*} \mathbb{F}\{\!\{t\}\!\} = \bigcup_{n \geq 1} \mathbb{F}(\!(t^{\frac{1}{n}})\!) = \left\{\left.\sum_{i=0}^\infty c_it^{g_i}\vphantom{\begin{array}{c} c_i \in \mathbb{F} \, , \, c_0 \neq 0 \\ g_i \in \mathbb{Q} \text{ with common denominator }\\g_0 \lt g_1 \lt \dots \lt g_i \lt \dots \end{array} }\ \right|\ \begin{array}{c} c_i \in \mathbb{F} \, , \, c_0 \neq 0 \\ g_i \in \mathbb{Q} \text{ with common denominator }\\g_0 \lt g_1 \lt \dots \lt g_i \lt \dots \end{array} \vphantom{\sum_{i=0}^\infty c_it^{g_i}}\right\} \, , \end{equation*}

along with the zero element. If $\mathbb{F}$ is an algebraically closed field, then $\mathbb{F}\{\!\{t\}\!\}$ is also algebraically closed: in fact, it is the algebraic closure of the field of Laurent series $\mathbb{F}(\!(t)\!)$.

The ‘first’ term $c_0t^{g_0}$ is referred to as the leading term, where c ₀ is the leading coefficient and g ₀ is the leading exponent. The valuation on $\mathbb{F}\{\!\{t\}\!\}$ maps zero to $\infty$, and a non-zero series to its leading exponent:

\begin{align*} \operatorname{val}\colon \mathbb{F}\{\!\{t\}\!\} &\rightarrow \mathbb{Q} \cup\{\infty\} \\ \sum_{i=0}^\infty c_it^{g_i} &\mapsto g_0 \, . \end{align*}

Example 2.22. A more general example of a valued field is the field of Hahn series with value group Γ over a field $\mathbb{F}$. These are formal power series whose exponents are elements of the ordered abelian group Γ

\begin{equation*} \mathbb{F}[\![ t^{\Gamma} ]\!] = \left\{\left.\sum_{g \in G} c_g t^g\vphantom{c_g \in \mathbb{F}^\times \, , \, G \subseteq \Gamma \text{ well-ordered } }\ \right|\ c_g \in \mathbb{F}^\times \, , \, G \subseteq \Gamma \text{ well-ordered } \vphantom{\sum_{g \in G} c_g t^g}\right\} \, , \end{equation*}

along with the zero element. The field $\mathbb{F}[\![ t^{\Gamma} ]\!]$ is algebraically closed if $\mathbb{F}$ is algebraically closed and Γ is divisible i.e. for all $g \in \Gamma$ and $n \in \mathbb{N}$, there exists $h \in \Gamma$ such that $g = n \cdot h$.

We use the same terminology for leading term/coefficient/exponent as with Puiseux series: the condition that G is well-ordered ensures it exists. The valuation on $\mathbb{F}[\![ t^{\Gamma} ]\!]$ behaves identically to the valuation on $\mathbb{F}\{\!\{t\}\!\}$, mapping zero to $\infty$ and a non-zero series to its leading exponent:

(2.1)\begin{align} \operatorname{val}\colon \mathbb{F}[\![ t^{\Gamma} ]\!] &\rightarrow \Gamma \cup\{\infty\}\\ \sum_{g \in G} c_g t^g &\mapsto \gamma = \min(g \mid g \in G) \, .\nonumber \end{align}

Valuations can be considered hyperfield homomorphisms in the following way. By identifying $\Gamma \cup \{\infty\}$ with $\mathbb{K}\rtimes \Gamma$, where $\mathbb{0} = \infty$, we can instead consider $\operatorname{val}$ as a map to $\mathbb{K}\rtimes \Gamma$. The following proposition is easy to verify.

Proof. It is clear the first two conditions of a valuation are hyperfield homomorphism properties. Moreover, it is straightforward to check $\operatorname{val}(1) = 0$. For the final condition, recall that

\begin{equation*} \operatorname{val}(a) \boxplus \operatorname{val}(b) = \begin{cases} \min(\operatorname{val}(a), \operatorname{val}(b)) & \operatorname{val}(a) \neq \operatorname{val}(b) \\ \left\{\left.h \in \Gamma \cup \{\infty\}\vphantom{h \geq \operatorname{val}(a)}\ \right|\ h \geq \operatorname{val}(a)\vphantom{h \in \Gamma \cup \{\infty\}}\right\} & \operatorname{val}(a) = \operatorname{val}(b) \end{cases} \, . \end{equation*}

This implies that $\operatorname{val}(a+b) \in \operatorname{val}(a) \boxplus \operatorname{val}(b)$ is equivalent to the final valuation condition.

Proposition 2.23 shows that valuations are equivalent to homomorphisms from a field to the tropical extension $\mathbb{K}\rtimes \Gamma$. This motivates the study of enriched valuations: homomorphisms $\mathbb{F} \rightarrow \mathbb{H}\rtimes \Gamma$ from a field to a tropical extension, where $\mathbb{H}$ records additional information about elements over $\mathbb{F}$.

Example 2.24. For an ordered field $\mathbb{F}$ and ordered abelian group Γ, the valuation map (2.1) can be enriched to the signed valuation, which records the sign of a Hahn series:

\begin{align*} \operatorname{sval}\colon \mathbb{F}[\![ t^{\Gamma} ]\!] & \rightarrow \mathbb{S}\rtimes \Gamma ,\\ \sum_{g \in G} c_g t^g &\mapsto (\operatorname{sgn}(c_{\gamma}), \gamma) \, , \quad \gamma = \min(g \mid g \in G) \, , \end{align*}

where $\operatorname{sgn}$ defined as in example 2.17. Setting the value group Γ to $\mathbb{R}$ recovers the signed valuation to the signed tropical hyperfield $\mathbb{S} \rtimes \mathbb{R} \cong \mathbb{T}\mathbb{R}$.

Example 2.25. Consider the field of Hahn series $\mathbb{F}[\![ t^{\Gamma} ]\!]$ over an arbitrary field. We can enrich the usual valuation map (2.1) on $\mathbb{F}[\![ t^{\Gamma} ]\!]$ by defining the fine valuation $\operatorname{fval}$ that remembers the coefficient of its leading term, i.e.

\begin{align*} \operatorname{fval}\colon \mathbb{F}[\![ t^{\Gamma} ]\!] &\rightarrow \mathbb{F}\rtimes \Gamma \\ \sum_{g \in G} c_g t^g &\mapsto (c_\gamma , \gamma)\, , \quad \gamma = \min(g \mid g \in G) \, . \end{align*}

As an application of our results, the tropical geometry of this valuation is discussed in §5.

Example 2.26. Fixing $\mathbb{F} = \mathbb{C}$, the valuation map (2.1) can be enriched to the phased valuation, which records the phase or argument of a Hahn series:

\begin{align*} \operatorname{phval}\colon \mathbb{C}[\![ t^{\mathbb{R}} ]\!] & \rightarrow \mathbb{P}\rtimes \Gamma ,\\ \sum_{g \in G} c_g t^g &\mapsto (\operatorname{ph}(c_{\gamma}), \gamma)\, , \quad \gamma = \min(g \mid g \in G) \, , \end{align*}

where $\operatorname{ph}$ is the phase map as defined as in example 2.17.

We note that all of these examples also hold over Puiseux series, where the value group is $\Gamma = \mathbb{Q}$. However, we also note that $\mathbb{F}\{\!\{t\}\!\} \subsetneq\mathbb{F}[\![ t^{\mathbb{Q}} ]\!]$ as the latter has no common denominator condition.

We end this section by noting that enriched valuations allow us to construct these hyperfields as factor hyperfields. Explicitly, in each case, we can realize each hyperfield as the domain of the enriched valuation modulo the preimage of $\mathbb{1}$ in the hyperfield:

\begin{align*} \mathbb{T} &\cong \mathbb{F}[\![ t^{\mathbb{R}} ]\!]/\operatorname{val}^{-1}(0) & \mathbb{T}\mathbb{R} &\cong \mathbb{F}[\![ t^{\mathbb{R}} ]\!]/\operatorname{sval}^{-1}((\mathbb{1}_{\mathbb{S}},0)) \\ \mathbb{F} \rtimes \Gamma &\cong \mathbb{F}[\![ t^{\Gamma} ]\!]/\operatorname{fval}^{-1}((\mathbb{1}_{\mathbb{F}},0)) & \mathbb{P} \rtimes \Gamma &\cong \mathbb{C}[\![ t^{\Gamma} ]\!]/\operatorname{phval}^{-1}((\mathbb{1}_{\mathbb{P}},0)) \\ \end{align*}

The proof of these isomorphisms follows from the following lemma.

Lemma 2.27. Let $f\colon \mathbb{F} \rightarrow \mathbb{H}$ be a surjective homomorphism from a field $\mathbb{F}$ to a hyperfield $\mathbb{H}$ satisfying

(2.2)\begin{equation} \forall \alpha, \beta \in \mathbb{H} \, , \, \gamma \in \alpha \boxplus \beta \, , \, \exists a \in f^{-1}(\alpha) \, , \, b \in f^{-1}(\beta) \text{ such that } a+b \in f^{-1}(\gamma) \, . \end{equation}

Then $\mathbb{H} \cong \mathbb{F}/f^{-1}(\mathbb{1}_{\mathbb{H}})$.

Proof. Denote $U = f^{-1}(\mathbb{1}_{\mathbb{H}})$ and consider the map $f_U \colon \mathbb{F}/U \rightarrow \mathbb{H}$ that maps $\mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu \mapsto f(a)$. To see this map is well-defined, consider $a,b \in \mathbb{F}$ such that $\mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu =\mkern 1.5mu\overline{\mkern-2.5mub\mkern-1.5mu}\mkern 2.5mu$, i.e. there exists $u \in U$ such that $a = b \cdot u$. Then

\begin{equation*} f_U(\mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu) = f(a) = f(b \cdot u) = f(b) \odot f(u) = f(b) \odot \mathbb{1}_{\mathbb{H}} = f(b) = f_U(\mkern 1.5mu\overline{\mkern-2.5mub\mkern-1.5mu}\mkern 2.5mu) \, , \end{equation*}

hence the map f_U is independent of the choice of representative for the cosets.

We claim that f_U defines an isomorphism. It is straightforward to check that f_U is a surjective homomorphism given that f is. To see that f_U is injective, suppose that $f(a) = f(b)$. Then

\begin{equation*} f(a) \cdot f(b)^{-1} = f(a \cdot b^{-1}) = \mathbb{1}_{\mathbb{H}} \, \Rightarrow \, a \cdot b^{-1} \in U \, . \end{equation*}

Hence there exists $u \in U$ such that $a = b \cdot u$, and so $\mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu = \mkern 1.5mu\overline{\mkern-2.5mub\mkern-1.5mu}\mkern 2.5mu$.

As f_U is a bijective morphism, there exists an inverse map $f_U^{-1}\colon \mathbb{H} \rightarrow \mathbb{F}/U$ given by $\alpha \mapsto \mkern 1.5mu\overline{\mkern-2.5muf^{-1}(\alpha)\mkern-1.5mu}\mkern 2.5mu$. It remains to show this is also a homomorphism. The only non-trivial condition is that it preserves sums. Consider some $\gamma \in \alpha \boxplus \beta$, then $f^{-1}_U(\gamma) = \mkern 1.5mu\overline{\mkern-2.5muf^{-1}(\gamma)\mkern-1.5mu}\mkern 2.5mu$. By condition (2.2), there exists $a \in f^{-1}(\alpha)$ and $b\in f^{-1}(\beta)$ such that $a+b \in f^{-1}(\gamma)$. Coupled with the injectivity of $f_U^{-1}$, this implies

\begin{equation*} \mkern 1.5mu\overline{\mkern-2.5muf^{-1}(\gamma)\mkern-1.5mu}\mkern 2.5mu = \mkern 1.5mu\overline{\mkern-2.5mua+b\mkern-1.5mu}\mkern 2.5mu \subseteq \mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu+\mkern 1.5mu\overline{\mkern-2.5mub\mkern-1.5mu}\mkern 2.5mu \subseteq f_U^{-1}(\alpha) \boxplus f^{-1}_U(\beta) \, . \end{equation*}

Ranging over all $\gamma \in \alpha \boxplus \beta$ gives $f_U^{-1}(\alpha \boxplus \beta) \subseteq f_U^{-1}(\alpha) \boxplus f^{-1}_U(\beta)$.

Remark 2.28. Without condition (2.2), lemma 2.27 gives a bijective homomorphism from $\mathbb{F}/f^{-1}(\mathbb{1}_{\mathbb{H}})$ to $\mathbb{H}$. It is important to stress that this is not sufficient for an isomorphism of hyperfields, as the following example highlights.

Let $\mathbb{W} = \{\mathbb{1}, \mathbb{0}, -\mathbb{1}\}$ be the weak hyperfield of signs, with the same multiplication and addition as $\mathbb{S}$ aside from

\begin{equation*} \mathbb{1} \boxplus \mathbb{1} = -\mathbb{1} \boxplus - \mathbb{1} = \{\mathbb{1}, - \mathbb{1}\} \, . \end{equation*}

One can check that the sign map $\operatorname{sgn}\colon \mathbb{R} \rightarrow \mathbb{W}$ is also a surjective homomorphism, but $\mathbb{R}/\operatorname{sgn}^{-1}(\mathbb{1}) \cong \mathbb{S} \not\cong \mathbb{W}$. This is because $\operatorname{sgn}\colon \mathbb{R} \rightarrow \mathbb{W}$ does not satisfy (2.2): we have $-\mathbb{1} \in \mathbb{1} \boxplus \mathbb{1}$ but we cannot find positive reals $a, b$ such that a + b is negative.

2.2 Polynomials over hyperfields

We review some key notions of polynomials over hyperfields. Some care is needed, as we shall see that polynomials behave much worse over hyperfields than over fields.

Definition 2.29. The set of polynomials in n variables over ahyperfield $\mathbb{H}$ will be denoted $\mathbb{H}[X_1 \, , \dots , \, X_n]$, where elements of this set are defined as

(2.29)\begin{equation} p(X_1 \, , \dots , \, X_n) := {\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}} \odot X_1^{d_1} \odot \, \cdots \, \odot X_n^{d_n} \, , \end{equation}

where $D \subset \mathbb{Z}_{\geq 0}^n$ finite and $c_{\boldsymbol{d}} \in \mathbb{H}^\times$. Each polynomial defines a function from $\mathbb{H}^n$ to $\mathcal{P}^*(\mathbb{H})$ given by evaluation.

The set of Laurent polynomials over $\mathbb{H}$ will be denoted $\mathbb{H}[X_1^\pm, \dots, X_d^\pm]$, whose elements are polynomials whose exponents $D \subset \mathbb{Z}^n$ may take negative values. Each Laurent polynomial defines a function from $(\mathbb{H}^\times)^n$ to $\mathcal{P}^*(\mathbb{H})$, as $\mathbb{0}^{k}$ is undefined for negative values of k.

We will use multi-index notation $\boldsymbol{X}^{\boldsymbol{d}} = X_1^{d_1}\odot\cdots\odot X_n^{d_n}$ to denote (Laurent) monomials.

Definition 2.30. Let $p = {\Large\boxplus}_{\boldsymbol{d}\in D} c_{\boldsymbol{d}} \odot \boldsymbol{X}^{\boldsymbol{d}} \in \mathbb{H}[X_1,\dots,X_n]$. An element $\boldsymbol{a} = (a_1 \, , \dots , \, a_n)$ is a root of the polynomial if $\mathbb{0} \in p(\boldsymbol{a}) = {\Large\boxplus}_{\boldsymbol{d}\in D} c_{\boldsymbol{d}} \odot {\boldsymbol{a}}^{\boldsymbol{d}}$. We denote the set of roots of p as

\begin{equation*} V(p) : = \{\boldsymbol{a} \in \mathbb{H}^n \mid \mathbb{0} \in p(\boldsymbol{a})\} \, . \end{equation*}

Note that V(p) can also be considered as the (affine) hypersurface defined by p. We will expand on this in §2.3.

Let $f:\mathbb{H}_1 \rightarrow \mathbb{H}_2$ be a hyperfield homomorphism. This induces a map of polynomials $f_* : \mathbb{H}_1[X_1,\dots,X_n] \rightarrow \mathbb{H}_2[X_1,\dots,X_n]$ defined as

\begin{equation*} p = {\Large\boxplus}_1 c_{\boldsymbol{d}} \odot_1 \boldsymbol{X}^{\boldsymbol{d}} \longmapsto f_*(p) = {\Large\boxplus}_2 f(c_{\boldsymbol{d}}) \odot_2 \boldsymbol{X}^{\boldsymbol{d}} \, . \end{equation*}

We call $f_*(p)$ the push-forward of p. By properties of hyperfield homomorphisms, we observe that

(2.2)\begin{equation} f(p(\boldsymbol{a})) = f\left({\Large\boxplus}_1 c_{\boldsymbol{d}} \odot_1 {\boldsymbol{a}}^{\boldsymbol{d}}\right) \subseteq {\Large\boxplus}_2 f(c_{\boldsymbol{d}}) \odot_2 (f(\boldsymbol{a}))^{\boldsymbol{d}} = f_*(p)(f(\boldsymbol{a})) \, . \end{equation}

This gives the following result as a direct consequence.

Lemma 2.31. [Reference Maxwell45]

Let $f: \mathbb{H}_1 \rightarrow \mathbb{H}_2$ be a hyperfield homomorphism. For $p \in \mathbb{H}_1[X_1 \, , \dots , \, X_n]$,

\begin{equation*} f(V(p)) \subseteq V(f_*(p)) \, . \end{equation*}

For general hyperfield homomorphisms, this containment is strict. However, for a particularly nice class of homomorphisms, we will be able to deduce an equality in lemma 2.31. These will be the class of RAC homomorphisms and will be the focus of §3.

2.3 Hyperfield varieties

We briefly touched on the notion of an affine hypersurface over $\mathbb{H}$ in the previous section. More general varieties are trickier as polynomial ideals over hyperfields have a number of pitfalls that we document at the end of this section. As such, we must restrict ourselves to images of polynomial ideals over fields. It will be useful for us to consider various notions of varieties over a hyperfield.

Affine varieties

Given a polynomial $p \in \mathbb{H}[X_1, \dots, X_n]$, its associated affine hypersurface is

\begin{equation*} V(p) := \left\{\left.\boldsymbol{a} \in \mathbb{H}^n\vphantom{\mathbb{0} \in p(\boldsymbol{a})}\ \right|\ \mathbb{0} \in p(\boldsymbol{a})\vphantom{\boldsymbol{a} \in \mathbb{H}^n}\right\} \, . \end{equation*}

Given a set of polynomials $J \subseteq \mathbb{H}[X_1, \dots, X_n]$, its associated affine prevariety is

\begin{equation*} V(J) := \left\{\left.\boldsymbol{a} \in \mathbb{H}^n\vphantom{\boldsymbol{a} \in V(p) \text{ for all } p \in J}\ \right|\ \boldsymbol{a} \in V(p) \text{ for all } p \in J\vphantom{\boldsymbol{a} \in \mathbb{H}^n}\right\} = \bigcap_{p \in J} V(p) \, . \end{equation*}

If there exists some ideal $I \subseteq \mathbb{F}[X_1,\dots,X_n]$ over a field $\mathbb{F}$ and homomorphism $f\colon \mathbb{F} \rightarrow \mathbb{H}$ such that $J = f_*(I) := \left\{\left.f_*(p)\vphantom{p \in I}\ \right|\ p \in I\vphantom{f_*(p)}\right\}$, we will call V(J) an affine variety.

Remark 2.32. We briefly note that there is a well-defined notion of a tropical variety that does not depend on any underlying ideal over a field. This differs from the notion above, where the varieties V(J) such that $J = \operatorname{val}_*(I)$ for some ideal I over a field $\mathbb{F}$ would be known as realizable tropical varieties. For general hyperfields, we do not have such intrinsic notions of varieties yet and so restrict ourselves to those that are ‘realizable’.

Projective varieties

As multiplication is single-valued, projective space and projective varieties over a hyperfield are defined much the same as over fields. As such, we shall skip over proofs of standard facts and refer to [Reference Cox, Little and O’Shea15], where proofs can be easily adapted to hyperfields. We define n-dimensional projective space over $\mathbb{H}$ as

\begin{equation*} \mathbb{P}^n(\mathbb{H}) = {\mathbb{H}^{n+1} \setminus \{\underline{\mathbb{0}}\}}\Big/{\sim} \, , \end{equation*}

where ∼ identifies scalar multiples, i.e. $\boldsymbol{a} \sim \lambda \odot \boldsymbol{a}$ for all $\lambda \in \mathbb{H}^\times$.

A polynomial $p \in \mathbb{H}[X_0, \dots, X_n]$ is homogeneous if each monomial has the same degree. If p is homogeneous, then $\boldsymbol{a} \in \mathbb{H}^{n+1}$ is a root if and only if $\lambda \odot \boldsymbol{a}$ is a root. As such, we can consider roots of homogeneous polynomials as hypersurfaces in projective space. Given a homogeneous polynomial $p \in \mathbb{H}[X_0, X_1, \dots, X_n]$, its associated projective hypersurface is

(2.3)\begin{align} PV(p) :=& \left\{\left.\boldsymbol{a} \in \mathbb{P}^n(\mathbb{H})\vphantom{\mathbb{0} \in p(\boldsymbol{a})}\ \right|\ \mathbb{0} \in p(\boldsymbol{a})\vphantom{\boldsymbol{a} \in \mathbb{P}^n(\mathbb{H})}\right\} = {V(p) \setminus \{\underline{\mathbb{0}}\}}\Big/{\sim} \, . \end{align}

Given a set of homogeneous polynomials $J \subseteq \mathbb{H}[X_0, X_1, \dots, X_n]$, its associated projective prevariety is

\begin{align*} PV(J) :=& \bigcap_{p \in J} PV(p) = {V(J) \setminus \{\underline{\mathbb{0}}\}}\Big/{\sim} \, . \end{align*}

If there exists an ideal $I \subseteq \mathbb{F}[X_0,X_1, \dots,X_n]$ over a field $\mathbb{F}$ and homomorphism $f\colon \mathbb{F} \rightarrow \mathbb{H}$ such that $J = f_*(I)$, we will call PV(J) a projective variety.

Torus subvarieties

Tropical geometry generally focuses on subvarieties of the torus, as these give rise to polyhedral complexes in $\mathbb{R}^n$. As such, we briefly recall the setup for this family of varieties also. Recall the n-dimensional torus over $\mathbb{H}$ is $(\mathbb{H}^\times)^n$. Given a Laurent polynomial $p \in \mathbb{H}[X_1^\pm, \dots, X_n^\pm]$, its associated torus hypersurface is

\begin{equation*} V^\times(p) := \left\{\left.\boldsymbol{a} \in (\mathbb{H}^\times)^n\vphantom{\mathbb{0} \in p(\boldsymbol{a})}\ \right|\ \mathbb{0} \in p(\boldsymbol{a})\vphantom{\boldsymbol{a} \in (\mathbb{H}^\times)^n}\right\} \, . \end{equation*}

Given a set of Laurent polynomials $J \subseteq \mathbb{H}[X_1^\pm, \dots, X_n^\pm]$, its associated torus prevariety is

\begin{equation*} V^\times(J) := \left\{\left.\boldsymbol{a} \in (\mathbb{H}^\times)^n\vphantom{\boldsymbol{a} \in V^\times(p) \text{ for all } p \in J}\ \right|\ \boldsymbol{a} \in V^\times(p) \text{ for all } p \in J\vphantom{\boldsymbol{a} \in (\mathbb{H}^\times)^n}\right\} = \bigcap_{p \in J} V^\times(p) \, . \end{equation*}

If $J = f_*(I)$ for some Laurent ideal $I \subseteq \mathbb{F}[X_1^\pm, \dots,X_n^\pm]$ over a field $\mathbb{F}$ and homomorphism $f\colon \mathbb{F} \rightarrow \mathbb{H}$, we will call $V^\times(J)$ a torus subvariety.

We will generally prove results for one of these three families of varieties and then deduce the analogous result for the other families. This requires a dictionary for how to move between these families. This is very similar to the situation over fields; therefore, we shall state results without proof.

Affine space, projective space and the torus are related by the following inclusions

\begin{equation*} \begin{array}{ccccc} (\mathbb{H}^\times)^n & \overset{i}{\hookrightarrow} & \mathbb{H}^n & \overset{j}\hookrightarrow & \mathbb{P}^n(\mathbb{H}) \\ (a_1, \dots a_n) & \mapsto & (a_1, \dots a_n) &\mapsto & [\mathbb{1} : a_1 : \dots : a_n] \end{array} \, . \end{equation*}

The first inclusion is canonical, and so we will drop the i. We denote $U = j(\mathbb{H}^n)$ and $U^\times = j((\mathbb{H}^\times)^n)$ as the affine and torus chart in $\mathbb{P}^n(\mathbb{H})$, respectively.

Lemma 2.33. Let $J \subseteq \mathbb{H}[X_1, \dots, X_n]$ be a collection of polynomials. The affine prevariety $V(J) \subseteq \mathbb{H}^n$ can be identified with the projective prevariety $PV(\mkern 1.5mu\overline{\mkern-2.5muJ\mkern-1.5mu}\mkern 2.5mu) \subseteq \mathbb{P}^n(\mathbb{H})$ restricted to the affine chart U, where $\mkern 1.5mu\overline{\mkern-2.5muJ\mkern-1.5mu}\mkern 2.5mu$ is the homogenization of J:

\begin{align*} \mkern 1.5mu\overline{\mkern-2.5muJ\mkern-1.5mu}\mkern 2.5mu &:= \left\{\left.{\Large\boxplus}_{d \in D} c_{d}\odot X_0^{\deg(p) - \sum_{i=1}^n d_i} \odot X_1^{d_1} \odot \cdots \odot X_n^{d_n}\vphantom{p = {\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}}\odot \boldsymbol{X}^{\boldsymbol{d}} \in J}\ \right|\ p = {\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}}\odot \boldsymbol{X}^{\boldsymbol{d}} \in J\vphantom{{\Large\boxplus}_{d \in D} c_{d}\odot X_0^{\deg(p) - \sum_{i=1}^n d_i} \odot X_1^{d_1} \odot \cdots \odot X_n^{d_n}}\right\}\\& \subseteq \mathbb{H}[X_0,X_1, \dots, X_n] \, . \end{align*}

Lemma 2.34. Let $J \subseteq \mathbb{H}[X_1^{\pm}, \dots, X_n^{\pm}]$ be a collection of Laurent polynomials. Then $V^\times(J) = V(J_{\operatorname{aff}}) \cap (\mathbb{H}^\times)^n$ where $J_{\operatorname{aff}}$ is the affinization of J:

\begin{align*} J_{\operatorname{aff}} &:= \left\{\left.{\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}}\odot X_1^{d_1+e_1} \odot \cdots \odot X_n^{d_n + e_n}\vphantom{{\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}}\odot \boldsymbol{X}^{\boldsymbol{d}} \in J \, , \, e_i = \min_{d \in D}(d_i, 0)}\ \right|\ {\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}}\odot \boldsymbol{X}^{\boldsymbol{d}} \in J \, , \, e_i = \min_{d \in D}(d_i, 0)\vphantom{{\Large\boxplus}_{\boldsymbol{d} \in D} c_{\boldsymbol{d}}\odot X_1^{d_1+e_1} \odot \cdots \odot X_n^{d_n + e_n}}\right\}\\& \subseteq \mathbb{H}[X_1, \dots, X_n] \, . \end{align*}

Moreover, the torus prevariety $V^\times(J) \subseteq (\mathbb{H}^\times)^n$ can be identified with the projective prevariety $PV(\overline{J_{\operatorname{aff}}})$ restricted to the torus chart $U^\times$.

We end this section with a compilation of issues with polynomial rings and ideals over hyperfields for algebraic geometry, and why we restrict ourselves to polynomial ideals coming from fields. This is not an exhaustive list: see [Reference Jun31, §4.1] for additional subtleties, as well as an alternative approach to hyperfield geometry.

Remark 2.35. Polynomial multiplication is multi-valued

The set of polynomials over a field $\mathbb{F}[X_1,\dots,X_n]$ has the structure of a ring. However, the set of polynomials over a general hyperfield $\mathbb{H}[X_1,\dots,X_n]$ is not a hyperring, as multiplication is not single-valued. Instead, both addition and multiplication are multi-valued, defined as

\begin{align*} \left({\Large\boxplus}_{\boldsymbol{d} \in D} a_{\boldsymbol{d}} \odot \boldsymbol{X}^{\boldsymbol{d}}\right) \boxplus \left({\Large\boxplus}_{\boldsymbol{d}' \in D'} b_{\boldsymbol{d}'} \odot \boldsymbol{X}^{\boldsymbol{d}'}\right) &= \Bigg\{{\Large\boxplus}_{\boldsymbol{e} \in D \cup D'} c_{\boldsymbol{e}} \odot X^{\boldsymbol{e}}\vphantom{c_{\boldsymbol{e}} \in a_{\boldsymbol{e}} \boxplus b_{\boldsymbol{e}}}\ \Bigg|\ c_{\boldsymbol{e}} \in a_{\boldsymbol{e}} \boxplus b_{\boldsymbol{e}}\vphantom{{\Large\boxplus}_{\boldsymbol{e} \in D \cup D'} c_{\boldsymbol{e}} \odot X^{\boldsymbol{e}}}\Bigg\} \, , \nonumber\\ \left({\Large\boxplus}_{\boldsymbol{d} \in D} a_{\boldsymbol{d}} \odot \boldsymbol{X}^{\boldsymbol{d}}\right) \boxdot \left({\Large\boxplus}_{\boldsymbol{d}' \in D'} b_{\boldsymbol{d}'} \odot \boldsymbol{X}^{\boldsymbol{d}'}\right) &= \Bigg\{{\Large\boxplus}_{\boldsymbol{e} \in D + D'} c_{\boldsymbol{e}} \odot X^{\boldsymbol{e}}\vphantom{c_{\boldsymbol{e}} \in {\Large\boxplus}_{\boldsymbol{e} = \boldsymbol{d} + \boldsymbol{d}'} a_{\boldsymbol{d}} \odot b_{\boldsymbol{d}'}}\ \Bigg| \\& \quad c_{\boldsymbol{e}} \in {\Large\boxplus}_{\boldsymbol{e} = \boldsymbol{d} + \boldsymbol{d}'} a_{\boldsymbol{d}} \odot b_{\boldsymbol{d}'}\vphantom{{\Large\boxplus}_{\boldsymbol{e} \in D + D'} c_{\boldsymbol{e}} \odot X^{\boldsymbol{e}}}\Bigg\} \, . \end{align*}

In the univariate case, it has the structure of a superring [Reference Ameri, Eyvazi and Hoskova-Mayerova7]. However even in the univariate case, this structure has a lot of discrepancies such as not being distributive [Reference Ameri, Eyvazi and Hoskova-Mayerova7], multiplication not being associative, and $\mathbb{H}[X_1, \dots, X_n]$ not being free in the sense of universal algebra [Reference Baker and Lorscheid12].

Remark 2.36. Roots of polynomials not closed under taking ideals

If we allow for multi-valued multiplication, we can still attempt to define an ideal as we usually would for polynomial rings. We define a hyperfield polynomial ideal $I \subseteq \mathbb{H}[X_1, \dots, X_n]$ to be a subset satisfying

1. (I1) $\mathbb{0} \in I$,

2. (I2) If $f \in I$, then $\lambda \boxdot f \subseteq I$ for all $\lambda \in \mathbb{H}[X_1, \dots, X_n]$,

3. (I3) If $f, g \in I$, then $f \boxplus g \subseteq I$.

It becomes quickly apparent that such a definition is too coarse for defining varieties over hyperfields.

Over a field, the common roots of a set of polynomials are also roots of all polynomials in the ideal they generate. This is not true for hyperfield polynomial ideals. Considering the additive closure in (I3), we may have two polynomials $f,g \in I$ with a common root $\boldsymbol{a} \in V(f) \cap V(g)$, but there exists some $h \in f \boxplus g \subseteq I$ such that $\boldsymbol{a} \notin V(h)$. This already occurs in simple examples: consider the univariate affine polynomials over $\mathbb{S}$:

\begin{equation*} f = X \boxplus \mathbb{1} \, , \quad g = -X \boxplus -\mathbb{1} \, , \quad h = X \boxplus -\mathbb{1} \in f \boxplus g \, . \end{equation*}

Both f and g have a unique root $X = -\mathbb{1}$, but the unique root of h is $X = \mathbb{1}$, despite $h \in f \boxplus g$. A similar phenomenon holds for the multiplicative action in (I2), where $f \in I$ has a root $\boldsymbol{a} \in V(f)$, but there exists some $\lambda \in \mathbb{H}[X_1, \dots, X_n]$ and $h \in \lambda \boxdot f$ such that $\boldsymbol{a} \notin V(h)$. For example, consider the univariate polynomials over $\mathbb{S}$:

\begin{equation*} f = X^2 \boxplus -X \boxplus \mathbb{1} \, , \quad h = X^3 \boxplus X^2 \boxplus X \boxplus \mathbb{1} \in (X \boxplus \mathbb{1}) \boxdot f \, . \end{equation*}

While f has the unique root $X = \mathbb{1}$, the unique root of h is $X = -\mathbb{1}$. Similar examples can easily be constructed for other hyperfields including $\mathbb{T}$.

Remark 2.37. Hyperfield polynomial ideals are too restrictive

In the case where $\mathbb{H} = \mathbb{T}$, we have a good notion of what tropical varieties and tropical ideals should be [Reference Maclagan and Rincón41]. We show that the only possible hyperfield polynomial ideals of $\mathbb{T}[X_1, \dots, X_n]$ are those generated by monomials, and hence are far too restrictive.

Let $f = {\Large\boxplus}_D c_{\boldsymbol{d}} \odot \boldsymbol{X}^{\boldsymbol{d}}$ be a generator for the polynomial ideal $I \subseteq \mathbb{T}[X_1, \dots, X_n]$. Then (I3) implies that

\begin{equation*} f \boxplus f = \left\{\left.{\Large\boxplus}_{\boldsymbol{d} \in D} a_{\boldsymbol{d}} \odot \boldsymbol{X}^{\boldsymbol{d}}\vphantom{a_{\boldsymbol{d}} \in [\infty, c_{\boldsymbol{d}}]}\ \right|\ a_{\boldsymbol{d}} \in [\infty, c_{\boldsymbol{d}}]\vphantom{{\Large\boxplus}_{\boldsymbol{d} \in D} a_{\boldsymbol{d}} \odot \boldsymbol{X}^{\boldsymbol{d}}}\right\} \subseteq I \, . \end{equation*}

In particular, the monomial $\boldsymbol{X}^{\boldsymbol{d}}$ is contained in I for all $\boldsymbol{d} \in D$. As we can write f as a linear combination of these monomials, we can replace f in the generating set by the monomials $\{\boldsymbol{X}^{\boldsymbol{d}} \mid \boldsymbol{d} \in D\}$. Repeating this over all generators, we obtain that I is generated by monomials. In particular, its associated variety $V(I) \subseteq \mathbb{T}^n$ can only be a collection of coordinate subspaces.

The situation is even worse when considering hyperfield Laurent polynomial ideals $I \subseteq \mathbb{T}[X_1^\pm, \dots, X_n^\pm]$, a common setup in tropical geometry. In this case, each of the generating monomials of I can be inverted and so I is either the zero ideal or all of $\mathbb{T}[X_1^\pm, \dots, X_n^\pm]$.

These examples demonstrate that hyperfield polynomial ideals are too large to be compatible with algebraic varieties. In contrast, collections of polynomials $f_*(J)$ coming from an ideal J over a field are smaller as they are not closed under addition and the multiplicative action of $\mathbb{H}[X_1, \dots, X_n]$, but their varieties are better behaved as we shall see in §4.2. It remains unclear what ‘ideal-like’ structure should be used for defining varieties over general hyperfields: we will briefly discuss this in §6.

3.1 RAC maps from fields

In the following, we observe that if $f \colon \mathbb{F} \rightarrow \mathbb{H}$ is a surjective homomorphism from an algebraically closed field to a stringent hyperfield, it is necessarily RAC. This result is deduced from a number of existing results in the literature.

We briefly recall the notion of multiplicity for roots of univariate polynomials over hyperfields. Recall from remark 2.35 that univariate polynomials have a hypermultiplication $\boxdot$ defined on them, giving $\mathbb{H}[X]$ the structure of a superring. Let $p \in \mathbb{H}[X]$, the multiplicity $\operatorname{mult}_a(p)$ of an element $a \in \mathbb{H}$ is defined as,

(3.1)\begin{equation} \operatorname{mult}_a(p) = \begin{cases} 0 & a \notin V(p) \\ 1 + \mathrm{max}\left\{\left.\operatorname{mult}_a(q)\vphantom{p(X) \in (X \boxplus -a) \boxdot q(X)}\ \right|\ p(X) \in (X \boxplus -a) \boxdot q(X)\vphantom{\operatorname{mult}_a(q)}\right\} & a \in V(p) \end{cases} \end{equation}

Due to multiplication of polynomials over $\mathbb{H}$ being multivalued, the polynomial q(X) in (3.1) is not necessarily unique. As such, the definition of multiplicity is necessarily recursive; see [Reference Baker and Lorscheid12] for details on the original definition and examples of the non-uniqueness.

Definition 3.10. A hyperfield $\mathbb{H}$ satisfies the multiplicity bound if for all univariate polynomials $p \in \mathbb{H}[X]$,

\begin{equation*} \sum_{a \in \mathbb{H}} \operatorname{mult}_a(p) \leq \deg(p) \, . \end{equation*}

If this inequality is an equality, we say $\mathbb{H}$ satisfies the multiplicity equality.

With this definition in hand, we make use of the following theorem that gives sufficient conditions for a map to be RAC.

This theorem is a special case of a more general statement given in [Reference Maxwell45], where one replaces algebraically closed field with a hyperfield that satisfies the multiplicity equality and the inheritance property. Informally, the inheritance property guarantees one can write a polynomial as a product of linear factors given by its roots with some other lower degree polynomial. Currently, this property is not well understood over hyperfields but is trivially satisfied by fields.

In comparison, multiplicities and the multiplicity bound have been much more studied for natural classes of hyperfields [Reference Akian, Gaubert and Tavakolipour5, Reference Gross and Gunn23, Reference Liu38]. In particular, we note the following result of Gunn:

Combining theorems 3.11 and 3.12 yields the following.

3.2 Tropical extensions of RAC maps

Given a homomorphism $f\colon \mathbb{H}_1 \rightarrow \mathbb{H}_2$, recall its tropical extension $f^\Gamma \colon \mathbb{H}_1\rtimes \Gamma\rightarrow \mathbb{H}_2\rtimes \Gamma$ introduced in definition 2.18. In this section, we prove that RAC maps are closed under taking tropical extensions.

To do so, we first note the following lemma that determines when an element is a root of a polynomial over a tropical extension.

Lemma 3.16. Let $p \in (\mathbb{H}\rtimes \Gamma)[X]$ and $a \in \mathbb{H}\rtimes \Gamma$ with $p(a) = {\Large\boxplus}_{i=0}^n (c_i,g_i)$, where $(c_i,g_i)$ is the evaluation of the monomial of degree i at a. Then a is a root of p if and only if there exists $J \subseteq \{0,1, \dots, n\}$ such that

\begin{align*} g_j \lt g_i \, , \, g_j = g_{j'} \quad \forall j, j' \in J, i \notin J \, , \qquad \text{ and } \qquad \mathbb{0}_{\mathbb{H}} \in {\Large\boxplus}_{j \in J}{}_{\mathbb{H}} \, c_j \, . \end{align*}

Proof of proposition 3.15

We will liberally use the conditions of lemma 3.16 throughout. Consider the polynomials

\begin{equation*} p = {\Large\boxplus}_{i=0}^n (\lambda_i,g_i) \odot X^{i} \in (\mathbb{H}_1\rtimes \Gamma)[X] , \quad f^\Gamma_*(p) = {\Large\boxplus}_{i=0}^n (f(\lambda_i),g_i) \odot X^{i} \in (\mathbb{H}_2\rtimes \Gamma)[X] \, . \end{equation*}

Let $(b,h) \in V(f^\Gamma_*(p))$ be a root of $f^\Gamma_*(p)$, and let $J \subseteq \{0, 1, \dots, n\}$ be the indices of the monomials at which the minimum is attained in Γ, i.e.

\begin{align*} g_j \odot h^{j} = g_j + j \cdot h \lt g_i + i \cdot h = g_i \odot h^{i} , \quad g_j \odot h^{j} = g_{j'} \odot h^{j'} , \quad \forall j, j' \in J \, , \, i \notin J \, . \end{align*}

Then b must be a non-zero root of the polynomial ${\Large\boxplus}_{j \in J} f(\lambda_j) \odot X^{j} \in \mathbb{H}_2[X]$. As f is RAC, there exists a non-zero root $a \in f^{-1}(b)$ of the polynomial ${\Large\boxplus}_{j \in J} \lambda_j \odot X^{j} \in \mathbb{H}_1[X]$. As $(a,h) \in \mathbb{H}_1\rtimes \Gamma$ satisfies the conditions of lemma 3.16, it is a root of p in the preimage of (b, h).

Recall from example 3.6 that a hyperfield is algebraically closed if the trivial homomorphism $\omega\colon \mathbb{H} \rightarrow \mathbb{K}$ is RAC. Applying proposition 3.15 gives us the following corollary.

Example 3.18. Recall the map $\eta(z) = -\log(|z|)$ from $\mathbb{T}\mathbb{C}$ to $\mathbb{T}$ introduced in example 2.19. It was shown in [Reference Maxwell45] that this map is RAC and was the only ‘non-trivial’ example known. Proposition 3.15 gives us a new perspective on this map via tropical extensions. Recall that we can view the map η as the tropical extension

\begin{align*} \omega^\Gamma \colon \Phi\rtimes \mathbb{R} &\rightarrow \mathbb{K}\rtimes \mathbb{R} \\ (\theta, g) &\mapsto (\mathbb{1},g) \, . \end{align*}

Example 3.3 shows that the tropical phase hyperfield is algebraically closed, hence corollary 3.17 shows η must also be RAC.

3.3 Quotients of RAC maps

In the following, we show that RAC maps are closed under ‘compatible’ quotients of the domain and target hyperfields.

Proposition 3.19. Let $f:\mathbb{H}_1 \rightarrow \mathbb{H}_2$ be an RAC homomorphism. If $U_1 \subseteq \mathbb{H}_1^{\times}$ and $U_2 \subseteq \mathbb{H}_2^{\times}$ such that $f(U_1) = U_2$, then the map

\begin{align*} \hat{f}: \mathbb{H}_1/U_1 & \rightarrow \mathbb{H}_2/U_2 \\ \mkern 1.5mu\overline{\mkern-2.5mua\mkern-1.5mu}\mkern 2.5mu & \mapsto \mkern 1.5mu\overline{\mkern-2.5muf(a)\mkern-1.5mu}\mkern 2.5mu \ \end{align*}

is RAC.

Proof. Let $\tau:\mathbb{H}_1 \rightarrow \mathbb{H}_1/U_1$ and $\sigma: \mathbb{H}_2 \rightarrow \mathbb{H}_2/U_2$ denote the corresponding quotient maps. Observe that $\hat{f}$ is defined such that the following diagram commutes,

i.e. we have $\hat{f}(\tau(a)) = \sigma(f(a))$ for all $a \in \mathbb{H}_1$. We show that $\hat{f}$ is well-defined similarly to the proof of lemma 2.27. Let $a,b \in \mathbb{H}_1$ such that $\tau(a) = \tau(b)$, i.e. there exists $u_1 \in U_1$ such that $a = b \odot_1 u_1$. Then,

\begin{align*} \hat{f}(\tau(a)) &= \sigma(f(a)) = \sigma(f(b \odot_1 u_1)) = \sigma(f(b) \odot_2 f(u_1)) \\ &= \sigma(f(b)) \odot_2 \sigma(f(u_1)) = \sigma(f(b)) \odot_2 \mathbb{1}_2 = \sigma(f(b)) = \hat{f}(\tau(b)), \end{align*}

as $f(u_1) \in U_2$. Thus, $\hat{f}$ is independent of the choice of representative for the coset. Moreover, it is straightforward to verify that $\hat{f}$ is also a surjective homomorphism.

To show it is RAC, let $Q \in (\mathbb{H}_1/U_1)[X]$ and $P = \hat{f}_*(Q) \in (\mathbb{H}_2/U_2)[X]$. We show that for some arbitrary root $z \in V(P)$, there exists $x \in V(Q)$ such that $\hat{f}(x) = z$. By [Reference Maxwell44, corollary 6.3.4], we can decompose the root set V(P) as follows

\begin{equation*} V(P) = \bigcup_{\sigma_*(p) = P} \sigma(V(p)) \, , \quad p \in \mathbb{H}_2[X] \, . \end{equation*}

This implies that there exists a polynomial $p \in \sigma_*^{-1}(P)$ with root $w \in V(p) \subseteq \mathbb{H}_2$ such that $\sigma(w) = z$. We now claim there exists a polynomial $q \in \tau_*^{-1}(Q) \subseteq \mathbb{H}_1[X]$ such that $f_*(q) = p$. To justify this, note that

\begin{equation*} \sigma_*(p) = P = \hat{f}_*(Q) = \hat{f}_*(\tau_*(q')) = \sigma_*(f_*(q')) \, , \end{equation*}

for all $q' \in \tau_*^{-1}(Q)$. Fixing some q ^ʹ, this implies that each coefficient of p differs from the corresponding coefficient of $f_*(q')$ by a scalar belonging to U ₂. As $f(U_1) = U_2$, there exists $q \in \mathbb{H}_1[X]$ such that $f_*(q) = p$ and has coefficients that differ from q ^ʹ by scalars in U ₁. Hence $\tau_*(q') = \tau_*(q)$, and so q satisfies the claim.

As f is RAC, for $w \in V(p)$ there exists $y \in f^{-1}(w)$ such that $y \in V(q)$. Applying lemma 2.31, we have

\begin{equation*} \tau(y) \in \tau(V(q)) \subseteq V(\tau_*(q)) = V(Q) \, . \end{equation*}

Setting $x = \tau(y)$, we see that

\begin{equation*} \hat{f}(x) = \hat{f}(\tau(y)) = \sigma(f(y)) = \sigma(w) = z \, , \end{equation*}

which concludes the proof.

4.1 Kapranov’s theorem

In this section, we prove a generalization of Kapranov’s theorem for RAC homomorphisms between hyperfields. We first prove it for affine hypersurfaces and show the projective and torus cases as corollaries.

Theorem 4.1 Let $f\colon \mathbb{H}_1 \rightarrow \mathbb{H}_2$ be an RAC homomorphism and $p \in \mathbb{H}_1[X_1, \dots, X_n]$ a polynomial. Then

\begin{equation*} V(f_*(p)) = f(V(p)) \, . \end{equation*}

The proof of this statement is very similar to the case where $\mathbb{H}_2 = \mathbb{T}$ given in [Reference Maxwell45], as we do not require $\mathbb{H}_2$ to have any additional properties. However, we include a proof for completeness.

Proof. Let $p = {\Large\boxplus}_{I \in \mathbb{Z}_{\geq 0}^n} c_I \odot \boldsymbol{X}^I \in \mathbb{H}[X_1, \dots, X_n]$ and pick some root $\boldsymbol{b} \in V(f_*(p))$ of the push-forward. Given lemma 2.31, it suffices to show there exists $\boldsymbol{a} \in V(p)$ such that $f(\boldsymbol{a}) = \boldsymbol{b}$.

Fix some $\lambda_i \in f^{-1}(b_i)$: note that such values exist as f is surjective. For any $D = (d_1, \dots, d_n) \in \mathbb{Z}_{\geq 0}^n$, we define the maps

\begin{align*} \phi_D\colon \mathbb{H}_1 &\rightarrow \mathbb{H}_1^n & \psi_D\colon \mathbb{H}_2 &\rightarrow \mathbb{H}_2^n \\ x &\mapsto (\lambda_1 \odot_1 x^{d_1}, \dots, \lambda_n \odot_1 x^{d_n}) & y &\mapsto (f(\lambda_1) \odot_2 y^{d_1}, \dots, f(\lambda_n) \odot_2 y^{d_n}) \, . \end{align*}

It is easy to verify by hyperfield homomorphism properties that the following diagram commutes.

Rather than pulling b back through the bottom row to find a root, we will instead traverse the other way around the square.

First note that $\psi_D(\mathbb{1}_2) = \boldsymbol{b}$. Consider $\phi_D^*(p) = p \circ \phi_D$, the pullback of the polynomial p through ϕ_D, i.e.

\begin{equation*} \phi_D^*(p) = {\Large\boxplus}_{I \in \mathbb{Z}_{\geq 0}^n} c_I \odot_1 (\lambda_1 \odot_1 X^{d_1})^{i_1} \odot_1 \cdots \odot_1 (\lambda_n \odot_1 X^{d_n})^{i_n} = {\Large\boxplus}_{I \in \mathbb{Z}_{\geq 0}^n} c_I \odot_1 \boldsymbol{\lambda}^I \odot_1 X^{D\cdot I} \, . \end{equation*}

Note that $\phi_D^*(p)$ may not be a polynomial in $\mathbb{H}_1[X]$, as we require coefficients to be elements rather than sets. There may exist two support vectors $I,I'$ of p such that $D\cdot I = D \cdot I'$, hence the hypersum of their coefficients may give a set. However, as p has finite support, we can choose D sufficiently generically such that $D\cdot I \neq D \cdot I'$ for all $I, I'$ in the support of p. This ensures $\phi_D^*(p) \in \mathbb{H}_1[X]$, as well as ensuring we cannot get any cancellation of terms.

By expanding out and using properties of homomorphisms, we see that

\begin{equation*} f_*(p)(\psi_D(X)) = f_*(\phi_D^*(p))(X) \quad \Rightarrow \quad \mathbb{0}_2 \in f_*(p)(\boldsymbol{b}) = f_*(\phi_D^*(p))(\mathbb{1}_2) \, . \end{equation*}

As f is an RAC homomorphism, there exists an element $\tilde{a} \in \mathbb{H}_1$ such that $\mathbb{0}_1 \in \phi_D^*(p)(\tilde{a})$ and $f(\tilde{a}) = \mathbb{1}_2$. Define $\boldsymbol{a} = \phi_D(\tilde{a})$, we then see that:

\begin{align*} p(\boldsymbol{a}) &= \phi_D^*(p)(\tilde{a}) \ni \mathbb{0}_1 \, , \\ f(\boldsymbol{a}) &= (f(\lambda_1)\odot_2 f(\tilde{a})^{d_1}, \dots, f(\lambda_n)\odot_2 f(\tilde{a})^{d_n}) = \boldsymbol{b} \, . \end{align*}

We can deduce a projective version of Kapranov’s theorem that follows immediately from theorem 4.1 and (2.3).

Corollary 4.2. Let $f\colon \mathbb{H}_1 \rightarrow \mathbb{H}_2$ be an RAC homomorphism and $p \in \mathbb{H}_1[X_0,X_1, \dots, X_n]$ a homogeneous polynomial. Then

\begin{equation*} PV(f_*(p)) = f(PV(p)) \, . \end{equation*}

We can also deduce a torus version of Kapranov’s theorem as a fairly immediate corollary.

Corollary 4.3. Let $f\colon \mathbb{H}_1 \rightarrow \mathbb{H}_2$ be an RAC homomorphism and $p \in \mathbb{H}_1[X_1^\pm, \dots, X_n^\pm]$ a Laurent polynomial. Then

\begin{equation*} V^\times(f_*(p)) = f(V^\times(p)) \, . \end{equation*}

Proof. We will utilize the projective case from corollary 4.2, restricted to the torus charts $U^\times_{\mathbb{H}_1}$ and $U^\times_{\mathbb{H}_2}$ using lemma 2.34. From this, we obtain

\begin{align*} f(V^\times(p))& = f(PV(\mkern 1.5mu\overline{\mkern-2.5mup_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu) \cap U^\times_{\mathbb{H}_1}) = f(PV(\mkern 1.5mu\overline{\mkern-2.5mup_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu)) \cap U^\times_{\mathbb{H}_2} = PV(f_*(\mkern 1.5mu\overline{\mkern-2.5mup_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu)) \cap U^\times_{\mathbb{H}_2}\nonumber\\ & = V^\times(f_*(p)) \, . \end{align*}

4.2 Extending the fundamental theorem of tropical geometry to hyperfields

In tropical geometry, Kapranov’s theorem can be extended to the fundamental theorem by considering varieties defined by polynomial ideals, rather than hypersurfaces defined by a single polynomial. The aim of this section is to present a parallel picture for RAC hyperfield homomorphisms. However, there are major deficiencies with polynomial ideals in the hyperfield setting as laid out in remarks 2.35, 2.36, and 2.37. As such, we restrict our setting to RAC maps from fields where polynomial ideals are well-behaved with respect to algebraic varieties.

Theorem 4.4 Let $f: \mathbb{F} \rightarrow \mathbb{H}$ be an RAC homomorphism from an algebraically closed field $\mathbb{F}$ to a hyperfield $\mathbb{H}$. Then, for any homogeneous ideal $I \subseteq \mathbb{F}[X_0,X_1 \dots X_n]$,

\begin{equation*} f(PV(I)) = PV(f_*(I)) \, . \end{equation*}

As with Kapranov’s theorem, one containment follows directly from hyperfield homomorphism properties and doesn’t require any (relatively) algebraically closed assumptions. Moreover, we will show the affine case, as the projective and torus versions of the statement follow immediately.

Lemma 4.5. Let $f: \mathbb{F} \rightarrow \mathbb{H}$ be a homomorphism from a field $\mathbb{F}$ to a hyperfield $\mathbb{H}$. For any ideal $I \subseteq \mathbb{F}[X_1, \dots X_n]$,

\begin{equation*} f(V(I)) \subseteq V(f_*(I)) \, . \end{equation*}

The other containment is more involved. We will use multiple facts from elementary algebraic geometry without proof: we refer to [Reference Cox, Little and O’Shea15] for further details. We will also make liberal use of lemma 2.31 and theorem 4.1.

Proof. The inclusion $f(PV(I)) \subseteq PV(f_*(I))$ follows from the affine case in lemma 4.5 and quotienting by scalars. We prove the reverse inclusion via induction on the dimension of the variety PV(I).

Let $\dim(PV(I)) = 0$, then $PV(I) = \{{\boldsymbol{a}}^{(1)}, \dots {\boldsymbol{a}}^{(l)} \} \subset \mathbb{P}^{n}(\mathbb{F})$ is a finite set of l points. Consider any $\boldsymbol{y} \in \mathbb{P}^{n}(\mathbb{H}) \setminus f(PV(I))$, we construct a polynomial in $f_*(I)$ that vanishes on $\{f({\boldsymbol{a}}^{(1)}), \dots , f({\boldsymbol{a}}^{(l)}) \}$ but not on y. For each $\boldsymbol{a} \in PV(I)$, fix $j \in \{0, \dots, n\}$ such that $f(a_j) \neq \mathbb{0}$. Then there exists $k \in \{0, \dots ,n \}\setminus j$ such that

\begin{equation*} \frac{f(a_k)}{f(a_j)} \neq \frac{y_k}{y_j} \quad \Rightarrow \quad \frac{a_k}{a_j} \neq \frac{w_k}{w_j} \quad \forall \boldsymbol{w} \in f^{-1}(\boldsymbol{y}) \subseteq \mathbb{P}^n(\mathbb{F}) \, . \end{equation*}

Note that these quantities are well-defined in projective space. Define the linear polynomial $P_{\boldsymbol{a}} = a_j \cdot X_k - a_k \cdot X_j$ and let $P = \prod_{\boldsymbol{a} \in PV(I)} P_{\boldsymbol{a}} \in \mathbb{F}[X_0, \dots, X_n]$. This is defined such that $P(\boldsymbol{a}) = 0$ for all $\boldsymbol{a} \in PV(I)$, but $P(\boldsymbol{w}) \neq 0$ for all $\boldsymbol{w} \in f^{-1}(\boldsymbol{y})$. By the Nullstellensatz, there exists $m \in \mathbb{N}$ such that $P^m \in I$ but still does not vanish on $f^{-1}(\boldsymbol{y})$. Applying theorem 4.1, we see that

\begin{align*} \boldsymbol{y} = f(\boldsymbol{w}) & \notin f(PV(P^m)) = PV(f_*(P^m)) \quad \Rightarrow \quad \boldsymbol{y} \notin \bigcap_{p \in I}PV(f_*(p)) = PV(f_*(I)) \, . \end{align*}

This completes the base case of $\dim(PV(I)) = 0$.

We now assume that the claim holds for all varieties of dimension less than k, and let $\dim(PV(I)) = k$. We will first prove the case where PV(I) is irreducible, i.e. I is prime, and consider the reducible case after.

Consider an arbitrary $\boldsymbol{y} \in \mathbb{P}^n(\mathbb{H}) \setminus f(PV(I))$ and fix some element $\boldsymbol{w} \in f^{-1}(\boldsymbol{y})$ in the preimage with corresponding maximal ideal $\mathfrak{m}_{\boldsymbol{w}} \subseteq \mathbb{F}[X_0, X_1,\dots, X_n]$. As $\mathfrak{m}_{\boldsymbol{w}}$ is maximal and $\dim(PV(I)) \gt 0$, there exists some $q \in \mathfrak{m}_{\boldsymbol{w}} \setminus I$. Geometrically, this is equivalent to $q(\boldsymbol{w}) = 0$ but q does not vanish on all of PV(I). By [Reference Cox, Little and O’Shea15, proposition 9.4.10], we have

\begin{equation*} \dim(PV(I) \cap PV(q)) = \dim(PV(I + \langle q\rangle) = k-1 \, . \end{equation*}

Applying the inductive hypothesis gives us

\begin{equation*} \boldsymbol{y} = f(\boldsymbol{w}) \notin f(PV(I)) \supseteq f(PV(I + \langle q\rangle)) = \bigcap_{p \in I +\langle q \rangle } PV(f_*(p)) \, . \end{equation*}

Hence, there exists $p \in I +\langle q \rangle$ such that $\mathbb{0} \notin f_*(p)(\boldsymbol{y})$, and therefore that $p(\boldsymbol{w}) \neq 0 $ for all $\boldsymbol{w}\in f^{-1}(\boldsymbol{y})$. By expressing p as a certain combination, we note

\begin{align*} p &= \hat{p} + \lambda q^m , \quad \hat{p} \in I \, , \, \lambda \in \mathbb{F}[X_0,\dots ,X_n] \, , \, m \in \mathbb{N} \\ \Rightarrow \, p(\boldsymbol{w}) &= \hat{p}(\boldsymbol{w}) + \lambda(\boldsymbol{w}) q^m(\boldsymbol{w}) = \hat{p}(\boldsymbol{w}) \neq 0 \, . \end{align*}

Hence, there exists a polynomial $\hat{p} \in I$ such that $\boldsymbol{w} \notin PV(\hat{p})$ for all $\boldsymbol{w} \in f^{-1}(\boldsymbol{y})$. Applying theorem 4.1 gives us

\begin{align*} \boldsymbol{y} & = f(\boldsymbol{w}) \notin f(PV(\hat{p})) = PV(f_*(\hat{p})) \quad \Rightarrow \quad \boldsymbol{y} \notin PV(f_*(\hat{p})) \supseteq \bigcap_{p \in I} PV(f_*(p))\\& = PV(f_*(I)) \, . \end{align*}

Finally, suppose that $PV(I) = \bigcup_{s=1}^r PV(I_s)$ is reducible into irreducible components $PV(I_s)$. On each irreducible component, we have $f(PV(I_s)) = PV(f_*(I_s))$. Therefore, applying lemma 4.6 gives us

\begin{equation*} f(PV(I)) = \bigcup_{s=1}^r f(PV(I_s)) = \bigcup_{s=1}^r PV(f_*(I_s)) = PV(f_*(I)) \, . \end{equation*}

Lemma 4.6. Let $f: \mathbb{F} \rightarrow \mathbb{H}$ be a relatively closed hyperfield homomorphism from an algebraically closed field $\mathbb{F}$ to a hyperfield $\mathbb{H}$. Let PV(I) be a projective variety with decomposition into irreducible components $PV(I) = PV(I_1) \cup \dots \cup PV(I_r)$. Then

\begin{equation*} PV(f_*(I)) = PV(f_*(I_1)) \cup \dots \cup PV(f_*(I_r)) \, . \end{equation*}

Proof. Assume $\boldsymbol{y} \in PV(f_*(I_s))$ for some $s \in [r]$, then $f_*(p)(\boldsymbol{y}) \ni \mathbb{0}$ for all $p \in I_s$. As $I \subseteq I_s$, it immediately follows $\boldsymbol{y} \in PV(f_*(I))$.

Conversely, if $\boldsymbol{y} \notin \bigcup_{s=1}^r PV(f_*(I_s))$, then for each $s \in [r]$ there exists some $p_s \in I_s$ such that $f_*(p_s)(\boldsymbol{y}) {\not\owns} \mathbb{0}$. This implies that $p_s(\boldsymbol{w}) \neq 0$ for all $\boldsymbol{w} \in f^{-1}(\boldsymbol{y})$. Let $p = \prod_{i=1}^r p_s \in \bigcap_{s=1}^r I_s$: this implies p is contained in the radical of I, and therefore, there exists $m \in \mathbb{N}$ such that $p^m \in I$. Moreover, $p^m(\boldsymbol{w}) \neq 0$ for all preimages w, and so applying Kapranov’s theorem gives

\begin{equation*} \boldsymbol{y} = f(\boldsymbol{w}) \notin f(PV(p^m)) = PV(f_*(p^m)) \supseteq PV(f_*(I)) \, . \end{equation*}

We get analogous statements for affine varieties and torus subvarieties by applying lemmas 2.33 and 2.34, respectively.

Corollary. Let $f: \mathbb{F} \rightarrow \mathbb{H}$ be an RAC homomorphism from an algebraically closed field $\mathbb{F}$ to a hyperfield $\mathbb{H}$. Then, for any ideal $I \subseteq \mathbb{F}[X_1 \dots X_n]$,

\begin{equation*} f(V(I)) = V(f_*(I)) \, . \end{equation*}

Moreover, for any Laurent ideal $J \subseteq \mathbb{F}[X_1^\pm, \dots, X_n^\pm]$,

\begin{equation*} f(V^\times(J)) = V^\times(f_*(J)) \, . \end{equation*}

Proof. We denote the affine chart over $\mathbb{F}$ and $\mathbb{H}$ as $U_{\mathbb{F}}$ and $U_{\mathbb{H}}$ respectively. By analogous arguments to those from corollary 4.3, observe that

\begin{equation*} f(PV(\mkern 1.5mu\overline{\mkern-2.5muI\mkern-1.5mu}\mkern 2.5mu) \cap U_{\mathbb{F}}) = f(PV(\mkern 1.5mu\overline{\mkern-2.5muI\mkern-1.5mu}\mkern 2.5mu)) \cap U_{\mathbb{H}} , \quad f(PV(\mkern 1.5mu\overline{\mkern-2.5muJ_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu) \cap U^\times_{\mathbb{F}}) = f(PV(\mkern 1.5mu\overline{\mkern-2.5muJ_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu)) \cap U^\times_{\mathbb{H}} \, . \end{equation*}

Combining lemma 2.33 and theorem 4.4 with this observation, we see

\begin{equation*} f(V(I)) = f(PV(\mkern 1.5mu\overline{\mkern-2.5muI\mkern-1.5mu}\mkern 2.5mu) \cap U_{\mathbb{F}}) = f(PV(\mkern 1.5mu\overline{\mkern-2.5muI\mkern-1.5mu}\mkern 2.5mu)) \cap U_{\mathbb{H}} = PV(f_*(\mkern 1.5mu\overline{\mkern-2.5muI\mkern-1.5mu}\mkern 2.5mu)) \cap U_{\mathbb{H}} = V(f_*(I)) \, . \end{equation*}

Similarly, combining lemma 2.34 and theorem 4.4 with the above observation gives the analogous statement for torus subvarieties,

\begin{align*} f(V^\times(J))& = f(PV(\mkern 1.5mu\overline{\mkern-2.5muJ_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu) \cap U^\times_{\mathbb{F}}) = f(PV(\mkern 1.5mu\overline{\mkern-2.5muJ_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu)) \cap U^\times_{\mathbb{H}}\\ & = PV(f_*(\mkern 1.5mu\overline{\mkern-2.5muJ_{\operatorname{aff}}\mkern-1.5mu}\mkern 2.5mu)) \cap U^\times_{\mathbb{H}} = V^\times(f_*(J)) \, . \end{align*}

5.1 Stable intersections

A fundamental aspect of tropical geometry is that the intersection of two tropical varieties may not be a tropical variety. Stable intersection turns out to be the correct notion of intersection, where one perturbs the tropical varieties before intersecting them. Formally, it is defined as

\begin{equation*} \operatorname{trop}(V(I)) \wedge \operatorname{trop}(V(J)) = \lim_{\epsilon \rightarrow 0} \left(\operatorname{trop}(V(I)) \cap (\operatorname{trop}(V(J)) + \epsilon\boldsymbol{v})\right) \end{equation*}

for some generic vector $\boldsymbol{v} \in \mathbb{R}^n$.

In [Reference Joswig and Smith30], a novel approach to stable intersection was introduced, where perturbation was performed in a higher rank tropical semiring, intersected set-theoretically and then projected to the usual rank one tropical semiring. In the following, we provide evidence that a similar paradigm holds by passing to a generic fine tropical variety.

Figure 2.

Left: The tropical hypersurfaces $\operatorname{trop}(V(P))$ and $\operatorname{trop}(V(Q))$ from example 5.8. Note that they intersect in a 1-dim ray, but they intersect stably at a single point. Right: The fine tropical hypersurfaces $\operatorname{ftrop}(V(P))$ and $\operatorname{ftrop}(V(Q))$ from the same example. These do intersect transversally, and so they intersect at a single point.

Example 5.8. Consider the bivariate polynomials

\begin{align*} P &= X + Y - 1 \, , & Q &= tX + (1+t^2) Y + 1 \, , & P,Q \in \mathbb{C}[\![ t^{\mathbb{R}} ]\!][X,Y] \, . \end{align*}

Their corresponding hypersurfaces $V(P), V(Q)$ are lines in the plane, and so their intersection is a single point, namely

\begin{align*} V(P) \cap V(Q) &= \left(\frac{2+t^2}{1-t+t^2}, \frac{-1-t}{1-t+t^2}\right)\\& = \left( 2 + 2t + t^2 + O(t^3), -1 -2t -t^2 + O(t^3) \right) \in \mathbb{C}[\![ t^{\mathbb{R}} ]\!]^2 \, . \end{align*}

Naively, we would expect the intersection of their tropical hypersurfaces to be the single point $(0,0)$. However, this is not the case as figure 2 demonstrates: their intersection is a one-dimensional ray, but their stable intersection is the correct point,

\begin{equation*} \operatorname{trop}(V(P)) \cap \operatorname{trop}(V(Q)) = \{(g, 0) \mid g \geq 0\} \, , \quad \operatorname{trop}(V(P)) \wedge \operatorname{trop}(V(Q)) = \{(0,0)\} \, . \end{equation*}

Instead let us consider their fine tropical hypersurfaces: $\operatorname{ftrop}(V(P))$ was computed in example 5.5, and $\operatorname{ftrop}(V(Q))$ is computed analogously:

\begin{align*} \operatorname{ftrop}(V(P)) &= \left\{\vphantom{ \begin{array}{l}g_X \gt g_Y = 0 \, , \, c_Y - 1 = 0 \\ g_Y \gt g_X = 0 \, , \, c_X - 1 = 0 \\ g_X = g_Y \lt 0 \, , \, c_X + c_Y = 0 \\ g_X = g_Y = 0 \, , \, c_X + c_Y - 1 = 0 \end{array}}\big((c_X,g_X) \; , \, (c_Y,g_Y)\big) \ \right. \\& \left.\left. \quad \quad \in (\mathbb{C} \rtimes \mathbb{R})^2\vphantom{ \begin{array}{l}g_X \gt g_Y = 0 \, , \, c_Y - 1 = 0 \\ g_Y \gt g_X = 0 \, , \, c_X - 1 = 0 \\ g_X = g_Y \lt 0 \, , \, c_X + c_Y = 0 \\ g_X = g_Y = 0 \, , \, c_X + c_Y - 1 = 0 \end{array}}\ \right| \begin{array}{l}g_X \gt g_Y = 0 \, , \, c_Y - 1 = 0 \\ g_Y \gt g_X = 0 \, , \, c_X - 1 = 0 \\ g_X = g_Y \lt 0 \, , \, c_X + c_Y = 0 \\ g_X = g_Y = 0 \, , \, c_X + c_Y - 1 = 0 \end{array}\vphantom{\big((c_X,g_X) \; , \, (c_Y,g_Y)\big) \in (\mathbb{C} \rtimes \mathbb{R})^2}\right\}\\ \operatorname{ftrop}(V(Q)) &= \left\{\vphantom{ \begin{array}{l} g_X \gt - 1 \, , \, g_Y = 0 \, , \, c_Y + 1 = 0 \\ g_X = -1 \, , \, g_Y \gt 0 \, , \, c_X + 1 = 0 \\ g_X + 1 = g_Y \lt 0 \, , \, c_X + c_Y = 0 \\ g_X +1 = g_Y = 0 \, , \, c_X + c_Y + 1 = 0 \end{array}} \big((c_X,g_X) \; , \, (c_Y,g_Y)\big) \right. \\& \left.\left. \quad \quad \in (\mathbb{C} \rtimes \mathbb{R})^2\vphantom{ \begin{array}{l} g_X \gt - 1 \, , \, g_Y = 0 \, , \, c_Y + 1 = 0 \\ g_X = -1 \, , \, g_Y \gt 0 \, , \, c_X + 1 = 0 \\ g_X + 1 = g_Y \lt 0 \, , \, c_X + c_Y = 0 \\ g_X +1 = g_Y = 0 \, , \, c_X + c_Y + 1 = 0 \end{array}}\ \right| \begin{array}{l} g_X \gt - 1 \, , \, g_Y = 0 \, , \, c_Y + 1 = 0 \\ g_X = -1 \, , \, g_Y \gt 0 \, , \, c_X + 1 = 0 \\ g_X + 1 = g_Y \lt 0 \, , \, c_X + c_Y = 0 \\ g_X +1 = g_Y = 0 \, , \, c_X + c_Y + 1 = 0 \end{array}\vphantom{\big((c_X,g_X) \; , \, (c_Y,g_Y)\big) \in (\mathbb{C} \rtimes \mathbb{R})^2}\right\} \end{align*}

It is not hard to check that these two hypersurfaces intersect at a single point $\big((2,0) \; , \, (-1,0)\big)$. Moreover, this point is precisely $\operatorname{fval}(V(P) \cap V(Q))$, and hence its image under the projection $\omega^{\mathbb{R}} \colon \mathbb{C} \rtimes \mathbb{R} \rightarrow \mathbb{K} \rtimes \mathbb{R} \cong \mathbb{T}$ is the stable intersection of $\operatorname{trop}(V(P))$ and $\operatorname{trop}(V(Q))$, i.e.

\begin{equation*} \operatorname{trop}(V(P)) \wedge \operatorname{trop}(V(Q)) = \omega^{\mathbb{R}}\left(\operatorname{ftrop}(V(P)) \cap \operatorname{ftrop}(V(Q))\right). \end{equation*}

Observe from figure 2 that the two rays that intersected in infinitely many points in $\mathbb{T}$ no longer intersect at all.

We note that moving to the fine tropical variety does not sidestep the need to perturb from all cases, as we may still encounter non-generic intersections. However, we do gain two advantages from this perspective, Firstly, non-generic intersections happen less frequently for fine tropical varieties as $\mathbb{C} \rtimes \mathbb{R}$ is in some sense a ‘bigger’ space than $\mathbb{T}$. Secondly, if we do need to perturb the fine tropical varieties, we expect it suffices to only perturb in $\mathbb{C}$ by sufficiently generic complex numbers, and leave the ‘tropical’ part as is. On the level of Hahn series, this corresponds to perturbing the leading coefficient while leaving the leading exponent be.

We can make this perturbation precise as follows. Let $P = \sum_{\boldsymbol{d} \in D} \rho_{\boldsymbol{d}} \cdot \boldsymbol{X}^{\boldsymbol{d}} \in \mathbb{C}[\![ t^{\mathbb{R}} ]\!][X_1^\pm, \dots, X_n^\pm]$ be a Laurent polynomial with support $D \subseteq \mathbb{Z}^n$, i.e. $\rho_{\boldsymbol{d}} \neq 0$ if and only if $\boldsymbol{d} \in D$. Given some $\alpha \in (\mathbb{C}^\times)^D$, we define the perturbed polynomial $P^\alpha = \sum_{\boldsymbol{d} \in D} \alpha_{\boldsymbol{d}} \cdot \rho_{\boldsymbol{d}} \cdot \boldsymbol{X}^{\boldsymbol{d}}$. Note that $\rho_{\boldsymbol{d}}$ and $\alpha_{\boldsymbol{d}} \cdot \rho_{\boldsymbol{d}}$ have the same leading exponents despite being different Hahn series. As such, the tropical varieties $\operatorname{trop}(V(P))$ and $\operatorname{trop}(V(P^\alpha))$ are equal, while their fine tropical varieties are distinct.

Conjecture 5.9.

Let $P, Q \in \mathbb{C}[\![ t^{\mathbb{R}} ]\!][X_1^\pm, \dots, X_n^\pm]$ where P has support $D \subseteq \mathbb{Z}^n$. Then there exists a Zariski open subset $A \subseteq (\mathbb{C}^\times)^D$ such that

\begin{equation*} \operatorname{trop}(V(P)) \wedge \operatorname{trop}(V(Q)) = \omega^{\mathbb{R}}(\operatorname{ftrop}(V(P^\alpha)) \cap \operatorname{ftrop}(V(Q))) \end{equation*}

for all $\alpha \in A$.

Example 5.10. Consider the polynomial $P' = X + Y + 1$, and recall the polynomial $Q = tX + (1+t^2) Y + 1$ from example 5.8. Despite P and P ^ʹ having the same tropical variety, they give rise to different fine tropical varieties. In particular, the fine tropical variety $\operatorname{ftrop}(V(P'))$ associated with P ^ʹ does not have generic intersection with $\operatorname{ftrop}(V(Q))$:

\begin{equation*} \operatorname{ftrop}(V(P')) \cap \operatorname{ftrop}(V(Q)) = \left\{\left.\big((c_X,g_X) \; , \, (-1,0)\big)\vphantom{g_X \gt 0 \, , \, c_X \in \mathbb{C}^\times}\ \right|\ g_X \gt 0 \, , \, c_X \in \mathbb{C}^\times\vphantom{\big((c_X,g_X) \; , \, (-1,0)\big)}\right\} \, . \end{equation*}

Moreover, the image of their intersection under $\omega^{\mathbb{R}}$ is not equal to the stable intersection $\operatorname{trop}(V(P')) \wedge \operatorname{trop}(V(Q))$.

By picking $\alpha = (1,1,-1) \in (\mathbb{C}^\times)^3$, we can perturb P ^ʹ to $(P')^\alpha = P$. It follows from example 5.8 that the intersection $\operatorname{ftrop}(V((P')^\alpha)) \cap \operatorname{ftrop}(V(Q))$ is generic, and that its image under $\omega^{\mathbb{R}}$ is equal to the stable intersection $\operatorname{trop}(V(P')) \wedge \operatorname{trop}(V(Q))$. However, conjecture 5.9 claims that almost any $\alpha \in (\mathbb{C}^\times)^3$ should suffice for this perturbation. Indeed, it can be checked that any $\alpha \in A$ gives the desired (stable) intersection, where

\begin{equation*} A = \left\{\left.(\alpha_X, \alpha_Y, \alpha_1) \in (\mathbb{C}^\times)^3\vphantom{\alpha_Y \neq \alpha_1}\ \right|\ \alpha_Y \neq \alpha_1\vphantom{(\alpha_X, \alpha_Y, \alpha_1) \in (\mathbb{C}^\times)^3}\right\} \, . \end{equation*}

5.2 Polyhedral homotopies

Polyhedral homotopy continuation is a method for finding solutions to systems of polynomial equations introduced by Huber and Sturmfels [Reference Huber and Sturmfels26]: we follow the tropical description in [Reference Leykin and Yu36]. Given a polynomial system $P = (p_1, \dots, p_n) \subset \mathbb{C}[X_1, \dots, X_n]$ with finitely many solutions, we wish to numerically approximate V(P). This is done by perturbing the coefficients of P by multiplying through by t ^γ for various generic $\gamma \in \mathbb{Q}$. This gives a perturbed system of equations over the Puiseux series $P_t = (p_1(t), \dots, p_n(t)) \subset \mathbb{C}\{\!\{t\}\!\}[X_1, \dots, X_n]$. If one can understand the solutions of P_t around $0 \lt t \ll 1$, then we can ‘track’ these solutions to an approximate solution of the original system $P = P_1$ via homotopy methods. We note that the method given in [Reference Leykin and Yu36] is more general than this, allowing us to find the solutions on a fixed algebraic variety, but we restrict to ordinary case for simplicity.

The method for finding an initial solution of P_t for small t is roughly as follows. By calculating the tropical variety $\operatorname{trop}(V(P_t))$, we obtain the leading exponents of the Puiseux series solutions. Moreover, for each $\boldsymbol{u} \in \operatorname{trop}(V(P_t))$, one can also recover the leading coefficient of the solution by computing the initial ideal $\operatorname{in}_{\boldsymbol{u}}(\langle P_t\rangle)$. Generically, the initial ideal should be zero-dimensional, and so the unique point $\boldsymbol{c} \in V^\times(\operatorname{in}_{\boldsymbol{u}}(\langle P_t\rangle))$ gives us a first-order approximation $(c_1t^{u_1}, \dots, c_nt^{u_n})$ of a solution in $V(P_t)$. Ranging over all $\boldsymbol{u} \in \operatorname{trop}(V(P_t))$ gives us a first-order approximation of our solution set $V(P_t)$ for t close to zero, which in general should be sufficient information to begin homotopy continuation.

By the characterization given in theorem 5.2, we can reframe this initial solution computation as computing the fine tropical variety $\operatorname{ftrop}(V(P_t))$. This motivates fine tropical varieties as a natural object of study when computing initial solutions for polyhedral homotopy continuation, and there are two avenues we hope this perspective will be useful. Firstly, computing the initial solution is a fairly expensive computation, requiring at least a mixed volume computation which is $\#P$-hard, if not a Gröbner basis computation which is worse-case doubly exponential. As such, any algorithmic advantages we can gain from the structure of fine tropical varieties, even in specific instances, would be welcome. Secondly, theorem 5.2 allows us to compute and encode initial solutions of polynomial systems with a positive dimensional solution set in only finite information. Explicitly, even if $\operatorname{trop}(V(P_t))$ has positive dimension, remark 5.4 shows that the initial ideal associated with each point are constant on cells. As such, we can describe the fine tropical variety $\operatorname{ftrop}(V(P_t))$ in finitely many polyhedral cells, each with a single initial ideal associated. We hope that this can be utilized alongside higher-dimensional continuation methods [Reference Sommese and Wampler50] to numerically approximate positive-dimensional varieties.