On intersections and stable intersections of tropical hypersurfaces

TL;DR

This paper proves that each connected component of the intersection of tropical hypersurfaces contains a point of their stable intersection unless the stable intersection is empty, linking algebraic and tropical geometry.

Contribution

It establishes a fundamental property of tropical hypersurface intersections, connecting algebraic hypersurfaces and their tropicalizations.

Findings

Connected components of tropical hypersurface intersections contain stable intersection points.

The result holds unless the stable intersection is empty.

Provides a bridge between algebraic and tropical geometry.

Abstract

We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless their stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize to them and the tropicalization of their intersection.

Equations

$$|{\Sigma }_1\wedge {\Sigma }_2|=\mathrm{trop}(X_1\cap t\cdot X_2)\text{for all }t\in U.$$

$$L\wedge \Sigma =\varnothing ⟺L^{\prime }\wedge \Sigma =\varnothing .$$

$$\Gamma \wedge {\Sigma }_{k+1}=\varnothing ⟺{\Gamma }^{\prime }\wedge {\Sigma }_{k+1}=\varnothing .$$

$$\phi :(K^n{)}^{k-1}\to K^{|A|},t\mapsto t^{-1}\cdot c.$$

Full text

Yue Ren is supported by UK Research and Innovation under the Future Leaders Fellowship programme (MR/S034463/2).

1. Introduction

Tropical varieties are commonly described as combinatorial shadows of algebraic varieties. The tropicalization of an algebraic variety shares many common properties with its algebraic counterparts, such as its dimension. This is prominently used in the finiteness proof of central configurations in the four and five body problem of celestial mechanics [HM06, HJ11], as well as finiteness proofs of many other central configurations such as those with fixed subconfigurations [HJ15] or equilateral chains [DH23]. In these proofs, the authors exploit the fact that their central configurations satisfy certain algebraic equations and thus lie on an algebraic variety. They then show that the tropicalization of the algebraic variety is zero-dimensional by intersecting the tropical hypersurfaces of carefully chosen equations and eliminating all resulting positive-dimensional polyhedra.

Computing an intersection of tropical hypersurfaces can however be an incredibly challenging task. Tropical hypersurfaces may have several maximal polyhedra and the computation requires intersecting all combinations thereof. While parallelisation and a clever choice of intersection order can lead to significant improvements in performance [JSV17], there is no general way to avoid the exponential number of intersections required. This circumstance is especially unsatisfying when one expects the final intersection to be very small, such as in all the cases above.

One alternative would be to compute the intersection “bottom-up”: if one can identify a point in every connected component of the intersection, the rest can be obtained using a traversal as in the computation of tropical varieties [BJSST07, MR20]. When the number of hypersurfaces equals the ambient dimension, a natural candidate for such a set of starting points is their stable intersection. This is in part due to similarity of definitions, see Figure 1, but also because it can be computed quickly using techniques such as tropical homotopy continuation [Jen16]. However, there is no known proof that every connected component of the intersection contains a point in the stable intersection. This paper aims to close that gap:

Theorem 3.3.

Let $\Sigma_{1},\dots\Sigma_{k}$ be tropical hypersurfaces in $\mathbb{R}^{n}$ with a non-empty stable intersection $\bigwedge_{i=1}^{k}\Sigma_{i}\neq\emptyset$. Then every connected component of their intersection $\bigcap_{i=1}^{k}|\Sigma_{i}|$ contains a point in the support of their stable intersection $|\bigwedge_{i=1}^{k}\Sigma_{i}|$.

Despite the obvious combinatorial nature of the statement, we were unable to proof the statement using combinatorics alone. Our proof relies on an algebro-geometric result by Josephine Yu on generic polynomials generating prime ideals [Yu16], and the properties of tropicalizations of irreducible varieties.

Acknowledgements

The work was partially done during Collaborate@ICERM “Numerical Algebraic Geometry and Tropical Geometry” with Tianran Chen (Auburn), Paul Helminck (Durham), Anders Jensen (Aarhus), Anton Leykin (Georgia Tech) and Josephine Yu (Georgia Tech). The author would like to thank them for helpful discussions, and the institute for its hospitality.

2. Background

In this article, we closely follow the notation of [MS15]. In particular:

Convention 2.1

For the remainder of the paper, we will fix an algebraically closed field $K$ of characteristic [math] with non-trivial valuation $\operatorname{val}\colon K^{\ast}\rightarrow\mathbb{R}$ and an element $t\in K^{\ast}$ with $\operatorname{val}(t)=1$. Let $K[x^{\pm 1}]\coloneqq K[x_{1}^{\pm 1},\dots,x_{n}^{\pm 1}]$ be a multivariate (Laurent) polynomial ring thereover.

For the sake of brevity, we will abbreviate “pure, weighted, balanced polyhedral complex” by “balanced polyhedral complex”, and we will consider tropical hypersurfaces as balanced polyhedral complexes instead of supports thereof. Additionally, we will denote the stable intersection by “$\wedge$” for better inline formatting.

We will further assume some familiarity with the basic concepts in Sections 2 and 3 of [MS15], such as the duality of tropical hypersurfaces and regular subdivisions of Newton polytopes, tropicalizations of algebraic varieties, stable intersections of balanced polyhedral complexes, and mixed volumes of polytopes. In particular, we will be relying heavily on the following results:

Theorem 2.2** **([MS15, Theorem 3.3.5] Structure Theorem for Tropical Varieties)

Let $X$ be an irreducible variety in $(K^{\ast})^{n}$ of dimension $d$. Then $\operatorname{trop}(X)$ is the support of a balanced polyhedral complex in $\mathbb{R}^{n}$ of dimension $d$ that is connected in codimension $1$.

Theorem 2.3** **([MS15, Theorem 3.6.1])

Let $\Sigma_{1},\Sigma_{2}$ be two balanced polyhedral complex in $\mathbb{R}^{n}$ whose support is the tropicalization of two varieties $X_{1},X_{2}\subseteq(K^{\ast})^{n}$. Then there is a Zariski dense subset $U\subseteq(K^{\ast})^{n}$ consisting of elements with component-wise valuation [math] such that

[TABLE]

Theorem 2.4** **([MS15, Theorem 3.6.10])

Let $\Sigma_{1},\Sigma_{2}$ be two balanced polyhedral complex in $\mathbb{R}^{n}$ of codimension $d,e$, respectively. If the stable intersection $\Sigma_{1}\wedge\Sigma_{2}$ is non-empty, then it is a balanced polyhedral complex of codimension $d+e$.

Additionally, we will require the following theorem by Yu. Note that [Yu16, Theorem 3] is more general and also covers fields of finite characteristic. We will only need and state its specialisation to fields of characteristic [math]. In the theorem, “general polynomials” means polynomials whose coefficients lie in a fixed Zariski open, dense set of the coefficient space.

Theorem 2.5** **([Yu16, Theorem 3])

Let $A_{1},\dots,A_{k}\subseteq\mathbb{Z}^{n}$, $k\leq n$, and $0\in A_{i}$ for all $i=1,\dots,k$. General polynomials in $K[x]$ with monomial supports $A_{1},\dots,A_{i}$ generate a proper ideal whose radical is prime if and only if for every $\emptyset\neq J\subseteq[k]$ one of the following holds:

1. (1)

$\dim\operatorname{Span}\bigcup_{j\in J}A_{j}>|J|$, or 2. (2)

$\dim\operatorname{Span}\bigcup_{j\in J}A_{j}=|J|$* and $\operatorname{MV}(\operatorname{Conv}(A_{j})\mid j\in J)=1$.*

3. Main Theorem

In this section, we prove Theorem 3.3, beginning with two lemmas. In the first lemma, we consider affine subspaces as balanced polyhedral complexes consisting of a single element. We show that if an affine subspace has an empty stable intersection with a balanced polyhedral complex, then so do its translates, see Figure 2.

Lemma 3.1

Let $\Sigma$ be a balanced polyhedral complex in $\mathbb{R}^{n}$. Let $L,L^{\prime}$ be two parallel affine subspaces in $\mathbb{R}^{n}$, i.e., $L=\mathrm{Ker}(A)+v$ and $L^{\prime}=\mathrm{Ker}(A)+v^{\prime}$ for some matrix $A\in\mathbb{R}^{k\times n}$ and vectors $v,v^{\prime}\in\mathbb{R}^{n}$. Then

[TABLE]

Proof.

Without loss of generality, we may assume that $L$ is a linear space, i.e., $0\in L$. Note that both $L$ and $L^{\prime}$ can be written as the supports of stable intersections of hyperplanes, i.e., $L=|H_{1}\wedge\dots\wedge H_{k}|$ and $L^{\prime}=|(H_{1}+v^{\prime})\wedge H_{2}\wedge\dots\wedge H_{k}|$ for suitable hyperplanes $H_{1},\dots,H_{k}$ and $v^{\prime}\in\mathbb{R}^{n}$. Since $H_{1}+v$ and $H_{1}+v^{\prime}$ remain parallel affine subspaces, and $H_{2}\wedge\dots\wedge H_{k}\wedge\Sigma$ remains a balanced polyhedral complex, we may assume without loss of generality that both $L$ and $L^{\prime}$ are affine hyperplanes.

Consider the projection of $|\Sigma|$ to the one-dimensional orthogonal complement $L^{\perp}$. The projection of $|\Sigma|$ remains the support of a balanced polyhedral complex, which means it is either a finite number of points or the entire complement. If the projection is a finite number of points, then $\Sigma$ is contained in a union of affine hyperplanes parallel to $L$ and $L^{\prime}$, and we have $L\wedge\Sigma=\emptyset=L^{\prime}\wedge\Sigma$. If the projection is the entire complement, then we have $L\wedge\Sigma\neq\emptyset\neq L^{\prime}\wedge\Sigma$. ∎

In the next lemma, we consider connected components of balanced polyhedral complexes as balanced polyhedral complexes. We use Lemma 3.1 to prove a similar statement but on the connected components of a stable intersection of tropical hypersurfaces instead of the translates of an affine subspace.

Lemma 3.2

Let $\Sigma_{1},\dots,\Sigma_{k},\Sigma_{k+1}$ be tropical hypersurfaces in $\mathbb{R}^{n}$ with $\bigwedge_{i=1}^{k}\Sigma_{i}\neq\emptyset$. Then for any two connected components $\Gamma,\Gamma^{\prime}\subseteq\bigwedge_{i=1}^{k}\Sigma_{i}$ we have

[TABLE]

Proof.

Let $A_{1},\dots,A_{k}\subseteq\mathbb{Z}^{n}$ be monomial supports of polynomials with tropical hypersurfaces $\Sigma_{1},\dots,\Sigma_{k}$ and $0\in A_{i}$ for all $i$. We distinguish between two cases:

**Case 1: **

If $A_{1},\dots,A_{k}$ do not satisfy Conditions (1) or (2) of Theorem 2.5, then there is a subset $J\subseteq[k]$ such that $\dim\operatorname{Span}(\bigcup_{j\in J}A_{j})\leq|J|$. Note that the lineality space of $\bigwedge_{j\in J}\Sigma_{j}$ contains the orthogonal complement of $\operatorname{Span}(\bigcup_{j\in J}A_{j})$. Therefore, the lineality space of $\bigwedge_{j\in J}\Sigma_{j}$ has codimension at most $|J|$. And since $\bigwedge_{j\in J}\Sigma_{j}\supseteq\bigwedge_{i=1}^{k}\Sigma_{i}\neq\emptyset$, Theorem 2.4 implies that $\bigwedge_{j\in J}\Sigma_{j}$ has codimension exactly $|J|$. Hence, $\bigwedge_{j\in J}\Sigma_{j}$ consists of affine subspaces parallel to its lineality space. The claim follows by applying Lemma 3.1 to $\Sigma\coloneqq\bigwedge_{i\in[k+1]\setminus J}\Sigma_{i}$ and the affine subspaces of $\bigwedge_{j\in J}\Sigma_{j}$.

**Case 2: **

If $A_{1},\dots,A_{k}$ satisfy Conditions (1) or (2) of Theorem 2.5, consider

• **•: **

$K\cdot x^{A}\coloneqq\bigoplus_{i=1}^{k}\bigoplus_{\alpha\in A_{i}}K\cdot x^{\alpha}$, the vector space of polynomial tuples $(f_{1},\dots,f_{k})$ where $f_{i}$ has monomial support $A_{i}$, and

• **•: **

$K^{|A|}\coloneqq K^{|A_{1}\cup\dots\cup A_{k}|}$, their coefficient space.

In particular, any choice of coefficients $c=(c_{i,\alpha})_{i\in[k],\alpha\in A_{i}}\in K^{|A|}$ defines a tuple of polynomials $f(c)\coloneqq(f_{i}(c))_{i\in[k]}\in K\cdot x^{A}$ with $f_{i}(c)\coloneqq\sum_{\alpha\in A_{i}}c_{i,\alpha}x^{\alpha}$ and vice versa. By Theorem 2.5, there is a Zariski open, dense subset $U\subseteq K^{|A|}$ such that any $c\in U$ yields an $f(c)$ whose entries generate an ideal whose radical is prime. By Theorem 2.2, the tropicalization $\operatorname{trop}(V(f(c)))\coloneqq\operatorname{trop}(V(\langle f_{1}(c),\dots,f_{k}(c)\rangle))$ will be connected for any $c\in U$. We will now show that there is a $c_{0}\in U$ such that $\operatorname{trop}(V(f(c_{0})))=|\bigwedge_{i=1}^{k}\Sigma_{i}|$, so that the claim holds trivially as there is only one connected component.

Pick any $c\in U$. By Theorem 2.3, there is a Zariski dense $S\subseteq(K^{n})^{k-1}$ with $\operatorname{trop}(V(f_{1}(c))\cap t_{2}V(f_{2}(c))\cap\cdots\cap t_{k}V(f_{k}(c)))=|\bigwedge_{i=1}^{k}\Sigma_{i}|$ for all ${(t_{2},\dots,t_{k})\in S}$. For any $t=(t_{2},\dots,t_{k})\in(K^{n})^{k-1}$, let $t^{-1}\cdot c\in K^{|A|}$ denote the coefficients of the polynomials $f_{1}(t^{-1}\cdot c)=\sum_{\alpha\in A_{1}}c_{1,\alpha}x^{\alpha}$ and $f_{i}(t^{-1}\cdot c)=\sum_{\alpha\in A_{i}}c_{i,\alpha}t_{i}^{-\alpha}x^{\alpha}$ for $i>1$, so that $V(f_{1}(t^{-1}c))=V(f_{1}(c))$ and $V(f_{i}(t^{-1}c))=t_{i}V(f_{i}(c))$ for $i>1$. Consider the polynomial map

[TABLE]

The preimage $\varphi^{-1}(U)$ is a Zariski open set, and since $\varphi(1,\dots,1)=c\in U$ it is non-empty. Thus $\varphi^{-1}(U)$ has to intersect the Zariski dense set $S$, and any point in the image of the intersection gives us the desired $c_{0}\in U$. ∎

Using Lemma 3.2 we can now prove:

Theorem 3.3

Let $\Sigma_{1},\dots\Sigma_{k}$ be tropical hypersurfaces in $\mathbb{R}^{n}$ with a non-empty stable intersection $\bigwedge_{i=1}^{k}\Sigma_{i}\neq\emptyset$. Then every connected component of their intersection $\bigcap_{i=1}^{k}|\Sigma_{i}|$ contains a point in the support of their stable intersection $|\bigwedge_{i=1}^{k}\Sigma_{i}|$.

Proof.

Let $C$ be a connected component of $\bigcap_{i=1}^{k}|\Sigma_{i}|$ and assume that $C$ contains no point of the stable intersection $\bigwedge_{i=1}^{k}\Sigma_{i}$. As $C$ contains a point of the stable intersection $\bigwedge_{j^{\prime}\in\{j\}}\Sigma_{j^{\prime}}=\Sigma_{j}$ for any singleton $\{j\}\subseteq[n]$, there is a maximal subset $J\subsetneq[k]$ such that $C$ contains a point of the stable intersection $\bigwedge_{j\in J}\Sigma_{j}$. Let $\Gamma\subseteq|\bigwedge_{j\in J}\Sigma_{j}|$ be the connected component containing said point. Note that $\Gamma$ is positive dimensional by Theorem 2.4.

If $\Gamma$ is completely contained in $C$, then $\Gamma\wedge\Sigma_{i}=\emptyset$ for any $i\notin J$ due to the maximality of $J$. Lemma 3.2 then implies $\bigwedge_{i=1}^{k}\Sigma_{i}=\emptyset$, contradicting the assumptions.

If $\Gamma$ is not completely contained in $C$, then, then there is a point $w\in C\cap\Gamma$ and a direction $u\in\mathbb{R}^{n}$ such that $w+\varepsilon\cdot u\in\Gamma$ but $w+\varepsilon\cdot u\notin C$ for $\varepsilon>0$ sufficiently small, see Figure 3 left. Since $w+\varepsilon\cdot u\notin C$, we also have $w+\varepsilon\cdot u\notin\bigcap_{i=1}^{k}|\Sigma_{i}|$. But because $w+\varepsilon\cdot u\in\bigcap_{j\in J}|\Sigma_{j}|$, there must be an $i_{0}\notin J$ such that $w+\varepsilon\cdot u\notin|\Sigma_{i}|$. Let $\gamma\in\Gamma$ be a polyhedron containing $w+\varepsilon\cdot u$. Let $\sigma\in\Sigma_{i_{0}}$ be a maximal polyhedron on the boundary of the region of $\mathbb{R}^{n}\setminus|\Sigma_{i_{0}}|$ containing $w+\varepsilon\cdot u$, see Figure 3 right. Then $\dim(\gamma+\sigma)=n$ and hence $w\in\gamma\cap\sigma\in\Gamma\wedge\Sigma_{i_{0}}$ by the definition of stable intersection. As $w\in C$, this contradicts that $J\subseteq[n]$ is maximal. ∎

4. Open questions

Recall that our approach to prove Theorem 3.3 relies on the fact that each $\Sigma_{i}$ is the tropicalization of an algebraic variety and on dimension arguments to show (non-)emptiness. Consequently, it has two key limitations:

First, it cannot deal with balanced polyhedral complexes which do not arise as tropicalizations of algebraic varieties. Such balanced polyhedral complexes are known to exist. In fact, balanced polyhedral complexes that are tropicalizations of algebraic varieties exhibit particularly nice properties such as higher connectivity [MY21]. Hence it is not clear whether Theorem 3.3 generalises to arbitrary balanced polyhedral complexes:

Question 4.1

Let $\Sigma_{1},\dots,\Sigma_{k}$ be balanced polyhedral complexes in $\mathbb{R}^{n}$ with a non-empty stable intersection $\bigwedge_{i=1}^{k}\Sigma_{i}\neq\emptyset$. Does every connected component of their intersection $\bigcap_{i=1}^{n}|\Sigma_{i}|$ contain a point of their stable intersection $|\bigwedge_{i=1}^{k}\Sigma_{i}|$?

Second, it cannot predict where the stable intersection points lie. Recall that the original motivation was to find a way to identify a point on each connected component of an intersection of tropical hypersurfaces. In case the number of tropical hypersurfaces in $\mathbb{R}^{n}$ exceeds $n$, their stable intersection is empty by Theorem 2.4. Hence their stable intersection does not provide an easy way to construct such points. Instead, a natural alternative are the stable intersections of any $n$ tropiocal hypersurfaces, see Figure 4. However, since our techniques do not give any information on where those stable intersection points lie, we do not know where they lie on the intersection of all hypersurfaces:

Question 4.2

Let $\Sigma_{1},\dots,\Sigma_{k}$ be tropical hypersurfaces in $\mathbb{R}^{n}$ with $k>n$ and non-empty stable intersections $\bigwedge_{j\in J}\Sigma_{j}\neq\emptyset$ for all $J\in\binom{[k]}{n}$. Does every connected component of their intersection $\bigcap_{i=1}^{k}|\Sigma_{i}|$ contain a point of a stable intersection $|\bigwedge_{j\in J}\Sigma_{j}|$ for some $J\in\binom{[k]}{n}$?