Geometry

Tropical geometry

Replacing ordinary addition and multiplication with the tropical operations $\oplus=\min$ and $\otimes=+$ turns polynomial curves into piecewise-linear graphs, with roots reaching back to shortest-path algorithms and forward to open problems in enumerative and non-archimedean geometry.

IntuitionFrom ordinary arithmetic to tropical arithmetic

Imagine every "addition" means pick the cheaper of two prices, and every "multiplication" means add up the costs along a route. Define two new operations on numbers extended with $+\infty$ : tropical addition $a \oplus b = \min(a, b)$ and tropical multiplication $a \otimes b = a + b$ . A tropical polynomial is an ordinary polynomial with every $+$ read as $\oplus$ and every $\times$ read as $\otimes$ . Because $\otimes$ is just addition, a monomial such as $x^{\otimes 3}$ — three copies of $x$ multiplied tropically — becomes $3x$ , and the whole polynomial collapses into the minimum of finitely many linear functions. Instead of a smooth curve, its "zero set" becomes the set of points where that minimum is achieved by at least two of the linear pieces at once: a corner, or piecewise-linear graph.

Why "tropical"? The name honors Imre Simon (1943–2009), a Hungarian-born Brazilian computer scientist who pioneered the study of min-plus algebra in theoretical computer science starting in the late 1970s. His French colleagues in automata theory coined the adjective "tropical" as a friendly nod to Simon's home city of São Paulo — south of the Tropic of Capricorn — with, as Sturmfels and Speyer later put it, "no deeper meaning." The word stuck once mathematicians realized that doing algebraic geometry over the min-plus semiring produces a rich, genuinely geometric theory.

Example: Evaluating a tropical polynomial by hand

Take the one-variable tropical polynomial $p(x) = \min(2x + 3,\ x + 1,\ 2)$ , coming from the ordinary polynomial $2 \otimes x^{\otimes 2} \oplus 1 \otimes x \oplus 2$ (coefficients $2, 1, 2$ on $x^2, x, x^0$ ). Evaluate $p(1)$ .

Solution

Substitute $x = 1$ into each of the three linear pieces: $2(1) + 3 = 5$ , $1 + 1 = 2$ , and the constant piece $2$ . The tropical value is the ordinary minimum of these three numbers: $p(1) = \min(5, 2, 2) = 2$ . Notice the tie between the second and third pieces ( $2 = 2$ ) — this is exactly a corner point of the piecewise-linear graph, the tropical analogue of a root.

UndergraduateThe tropical semiring and tropical curves

Definition: Tropical semiring

The tropical semiring (in the min-plus convention) is the set $\mathbb{T} = \mathbb{R} \cup \{+\infty\}$ equipped with tropical addition $a \oplus b = \min(a, b)$ and tropical multiplication $a \otimes b = a + b$ . It is a semiring, not a ring: $\oplus$ and $\otimes$ are commutative, associative, and $\otimes$ distributes over $\oplus$ , but no element other than $+\infty$ has an additive inverse (there is no number $x$ with $\min(a, x) = +\infty$ for finite $a$ ). The additive identity is $+\infty$ (since $\min(a, +\infty) = a$ ) and the multiplicative identity is $0$ (since $a + 0 = a$ ). (A dual max-plus convention, with $\oplus = \max$ , is equally common in the literature; the two are related by $x \mapsto -x$ , and this page fixes the min-plus convention throughout.)

A tropical polynomial in $n$ variables is a finite $\oplus$ -sum of $\otimes$ -monomials, i.e. a function $p : \mathbb{R}^n \to \mathbb{R}$ of the form $p(x) = \bigoplus_{\alpha \in S} c_\alpha \otimes x^{\otimes \alpha} = \min_{\alpha \in S} \big(c_\alpha + \alpha \cdot x\big)$ , where $S$ is a finite set of exponent vectors $\alpha \in \mathbb{Z}^n$ and $c_\alpha \in \mathbb{R}$ . Every such $p$ is the pointwise minimum of finitely many affine-linear functions with integer slopes, hence piecewise linear and concave. The tropical hypersurface (for $n = 2$ , a tropical curve) cut out by $p$ is its corner locus $V(p) = \{x \in \mathbb{R}^n : \text{the minimum in } p(x) \text{ is attained by at least two terms}\}$ — precisely the set of points where $p$ fails to be linear, the tropical analogue of "where the polynomial vanishes."

p(x) = \bigoplus_{\alpha \in S} c_\alpha \otimes x^{\otimes \alpha} = \min_{\alpha \in S} \big(c_\alpha + \alpha \cdot x\big)

A tropical polynomial in $n$ variables is a finite $\oplus$ -sum of $\otimes$ -monomials, i.e. a function $p : \mathbb{R}^n \to \mathbb{R}$ of the form $p(x) = \bigoplus_{\alpha \in S} c_\alpha \otimes x^{\otimes \alpha} = \min_{\alpha \in S} \big(c_\alpha + \alpha \cdot x\big)$ , where $S$ is a finite set of exponent vectors $\alpha \in \mathbb{Z}^n$ and $c_\alpha \in \mathbb{R}$ . Every such $p$ is the pointwise minimum of finitely many affine-linear functions with integer slopes, hence piecewise linear and concave. The tropical hypersurface (for $n = 2$ , a tropical curve) cut out by $p$ is its corner locus, the set of points where at least two of the linear pieces tie for the minimum: $V(p) = \{x \in \mathbb{R}^n : |\{\alpha \in S : c_\alpha + \alpha \cdot x = p(x)\}| \ge 2\}$ — precisely the set of points where $p$ fails to be linear, the tropical analogue of "where the polynomial vanishes."

Example: The tropical line

Take the simplest degree- $1$ tropical polynomial in two variables with all coefficients $0$ : $q(x, y) = x \oplus y \oplus 0 = \min(x, y, 0)$ . Find its corner locus (the tropical curve it defines), and describe the three regions where each term wins.

Solution

The three linear pieces are $x$ , $y$ , and $0$ . Compare them pairwise: $x$ is the unique minimum when $x < y$ and $x < 0$ ; $y$ is the unique minimum when $y < x$ and $y < 0$ ; $0$ is the unique minimum when $x > 0$ and $y > 0$ . The corner locus — where two pieces tie — consists of exactly three rays from the origin: the ray $\{x = y \le 0\}$ (direction $(-1,-1)$ , where the $x$ - and $y$ -pieces tie), the ray $\{x = 0,\ y \ge 0\}$ (direction $(0,1)$ , where the $x$ -piece ties the constant), and the ray $\{y = 0,\ x \ge 0\}$ (direction $(1,0)$ , where the $y$ -piece ties the constant). This three-pronged figure — sometimes nicknamed the "Mercedes-Benz" curve — is the tropical line: the tropical analogue of an ordinary straight line.

\sum_{j=1}^{k} w_j v_j = 0

Theorem: The balancing condition

Let $p$ be a tropical polynomial in two variables and let $v$ be a vertex of its tropical curve $V(p)$ . Let the edges of $V(p)$ incident to $v$ have primitive integer direction vectors $v_1, \dots, v_k \in \mathbb{Z}^2$ (each pointing away from $v$ ) and positive integer weights $w_1, \dots, w_k$ . Then $\sum_{j=1}^{k} w_j v_j = 0$ .

Why is it true?

This is a conservation law, structurally identical to Kirchhoff's current law at a node of an electrical circuit: near $v$ , the polynomial $p$ is the minimum of the affine pieces that achieve equality at $v$ , and each edge is where exactly two of them stay tied. As you walk once around $v$ , the slope of $p$ jumps by an amount proportional to $w_j v_j$ each time you cross an edge; since $p$ is a single well-defined continuous function, these jumps must cancel out after a full turn, which is exactly the balancing equation. It is this local cancellation — not just any polyhedral complex will do — that makes a tropical curve genuinely algebraic, i.e. the corner locus of an honest tropical polynomial, rather than an arbitrary collection of rays and segments.

Proof

Recall that $p(x) = \min_{\alpha \in S}(c_\alpha + \alpha \cdot x)$ is dual to the regular subdivision $\Delta_p$ of its Newton polygon $\mathrm{conv}(S)$ obtained by lifting each point $\alpha \in S$ to height $c_\alpha$ and projecting the lower convex hull back down. Every edge $e$ of $V(p)$ is dual to an edge $e^*$ of $\Delta_p$ : $e$ is perpendicular to $e^*$ (rotate by $90^\circ$ ), and its weight $w$ equals the lattice length of $e^*$ . Every vertex $v$ of $V(p)$ is dual to a $2$ -dimensional cell (a polygon) $\sigma_v$ of $\Delta_p$ , and the edges of $V(p)$ incident to $v$ correspond, in matching cyclic order, exactly to the boundary edges of $\sigma_v$ . Traversing the boundary of the closed polygon $\sigma_v$ once around, its edge vectors $e_1^*, \dots, e_k^*$ sum to zero — a polygon returns to where it started: $e_1^* + \cdots + e_k^* = 0$ . Rotation by $90^\circ$ is a linear map $R$ , and each $R(e_j^*)$ equals $w_j v_j$ up to the fixed choice of orientation (the weight $w_j$ is the lattice length of $e_j^*$ , and $v_j$ is $e_j^*$ rotated and rescaled to a primitive vector). Applying the linear map $R$ to both sides of $e_1^* + \cdots + e_k^* = 0$ gives $w_1 v_1 + \cdots + w_k v_k = R(0) = 0$ , which is exactly the balancing condition.

Check the balancing condition on the tropical line above: at the origin, the three rays have primitive directions $(1,0)$ , $(0,1)$ , $(-1,-1)$ , each with weight $1$ (every coefficient of $q$ was $0$ , so every dual edge in the Newton triangle has lattice length $1$ ). Indeed $1\cdot(1,0) + 1\cdot(0,1) + 1\cdot(-1,-1) = (0,0)$ , confirming the theorem on this simplest possible example.

AdvancedFrom formal valuations to a genuinely algebraic theory

The tropical line above was built by hand from arbitrary coefficients, but the deepest tropical curves come from tropicalizing an actual algebraic variety. Let $K$ be a field with a valuation $\mathrm{val} : K^* \to \mathbb{R}$ — a function with $\mathrm{val}(ab) = \mathrm{val}(a) + \mathrm{val}(b)$ and $\mathrm{val}(a+b) \ge \min(\mathrm{val}(a), \mathrm{val}(b))$ — such as the field $\mathbb{C}\{\!\{t\}\}$ of Puiseux series (where $\mathrm{val}$ reads off the lowest exponent of $t$ ) or the $p$ -adic numbers $\mathbb{Q}_p$ (where $\mathrm{val}$ is the $p$ -adic valuation). Extend $\mathrm{val}$ coordinatewise to a map $\mathrm{trop} : (K^*)^n \to \mathbb{R}^n$ , $\mathrm{trop}(x_1, \dots, x_n) = (\mathrm{val}(x_1), \dots, \mathrm{val}(x_n))$ . For a variety $X \subseteq (K^*)^n$ , the tropicalization $\mathrm{trop}(X)$ is the closure of the image $\mathrm{trop}(X(K))$ in $\mathbb{R}^n$ .

The single most important fact tying this formal machinery back to genuine algebraic geometry is the Fundamental Theorem of Tropical Geometry: for a variety $X = V(I) \subseteq (K^*)^n$ cut out by an ideal $I$ over an algebraically closed field $K$ with a nontrivial valuation, $\mathrm{trop}(X)$ coincides with the set $V(\mathrm{trop}(I)))$ obtained by tropicalizing every polynomial in $I$ and intersecting their corner loci — so the analytic tropicalization (built from actual points of $X$ ) agrees with the purely combinatorial tropical variety built from the ideal alone. Mikhail Kapranov proved the hypersurface case (a single defining polynomial) in the 1990s; the general statement for arbitrary ideals, together with a full proof via Gröbner theory and initial ideals, is developed as the Fundamental Theorem in Diane Maclagan and Bernd Sturmfels's $2015$ textbook Introduction to Tropical Geometry, building on tropical Gröbner-basis techniques worked out earlier in the 2000s.

Long before anyone called any of this "tropical," computer scientists were already doing tropical linear algebra by hand. The all-pairs shortest path problem on a weighted directed graph is a textbook instance of $(\min, +)$ -algebra: if $A$ is the $n \times n$ matrix with $A_{ij}$ the weight of the edge from $i$ to $j$ (and $A_{ij} = +\infty$ when no edge exists, $A_{ii} = 0$ ), then tropical matrix multiplication is defined exactly as in ordinary linear algebra but with $\oplus$ replacing $+$ and $\otimes$ replacing $\times$ : $(A \otimes B)_{ij} = \bigoplus_k \big(A_{ik} \otimes B_{kj}\big) = \min_k \big(A_{ik} + B_{kj}\big)$ .

d_{ij}^{(k)} = \min\big(d_{ij}^{(k-1)},\ d_{ik}^{(k-1)} + d_{kj}^{(k-1)}\big)

Theorem: Shortest paths as a tropical matrix power

Let $A$ be the $(\min,+)$ weight matrix of a directed graph on $n$ vertices with no negative-weight cycles. Let $A^{\otimes m}$ denote the $m$ -fold tropical matrix product of $A$ with itself. Then the entry $(A^{\otimes(n-1)})_{ij}$ equals the length of the shortest path from $i$ to $j$ in the graph. This is exactly the quantity computed by the Floyd–Warshall algorithm, whose recurrence $d_{ij}^{(k)} = \min\big(d_{ij}^{(k-1)},\ d_{ik}^{(k-1)} + d_{kj}^{(k-1)}\big)$ builds up $A^{\otimes(n-1)}$ one intermediate vertex at a time.

Why is it true?

Entry $(A^{\otimes m})_{ij}$ tracks the length of the best walk from $i$ to $j$ using **at most $m$ edges**, because tropical matrix multiplication is exactly "combine a first leg and a rest-of-the-journey via addition, then keep only the cheapest combination" — precisely Bellman's principle of optimality for shortest paths. Iterating this $n - 1$ times is enough because a shortest simple path in an $n$ -vertex graph never needs more than $n - 1$ edges, so no further tropical squaring can improve the answer.

Proof

By induction on $m$ . Base case $m = 1$ : $(A^{\otimes 1})_{ij} = A_{ij}$ is trivially the length of the best walk using at most one edge. Inductive step: assume $(A^{\otimes m})_{ik}$ equals the length of the shortest walk from $i$ to $k$ using at most $m$ edges, for every $k$ . Then $(A^{\otimes(m+1)})_{ij} = (A^{\otimes m} \otimes A)_{ij} = \min_k\big((A^{\otimes m})_{ik} + A_{kj}\big)$ . Any walk from $i$ to $j$ with at most $m+1$ edges either uses at most $m$ edges already (covered by the $k = j$ , $A_{jj} = 0$ term) or splits as a walk of at most $m$ edges from $i$ to some vertex $k$ followed by one final edge $k \to j$ ; minimizing over all such $k$ recovers exactly $(A^{\otimes(m+1)})_{ij}$ , so the claim holds for $m + 1$ . Since there are no negative-weight cycles, no shortest walk needs to repeat a vertex, so every shortest path uses at most $n - 1$ edges; taking $m = n - 1$ shows $(A^{\otimes(n-1)})_{ij}$ is exactly the shortest-path distance, matching the Floyd–Warshall recurrence, which computes the same quantity by fixing intermediate vertices one at a time instead of edge-counts.

Example: A tiny tropical matrix squaring

Three vertices $A, B, C$ with directed edges $A \to B$ (weight $4$ ), $B \to C$ (weight $3$ ), and $A \to C$ (weight $9$ ). Its $(\min,+)$ weight matrix is $A = \begin{pmatrix} 0 & 4 & 9 \\ \infty & 0 & 3 \\ \infty & \infty & 0 \end{pmatrix}$ (rows/columns in order $A, B, C$ ). Compute $(A \otimes A)_{AC}$ and compare it to the direct edge weight $9$ .

Solution

By definition $(A \otimes A)_{AC} = \min_k(A_{Ak} + A_{kC}) = \min\big(A_{AA}+A_{AC},\ A_{AB}+A_{BC},\ A_{AC}+A_{CC}\big) = \min(0+9,\ 4+3,\ 9+0) = \min(9, 7, 9) = 7$ . The tropical squaring found the two-edge route $A \to B \to C$ of total weight $7$ , strictly cheaper than the direct edge of weight $9$ — exactly the shortest-path update that Floyd–Warshall performs when it considers $B$ as an intermediate vertex. Squaring once already suffices here since the shortest path uses only $2 \le n - 1 = 2$ edges.

This is the "s–t flow network" graph from the catalog, shown here (not for its usual max-flow role) simply as a concrete weighted directed graph — the literal historical setting where

(\min,+)

matrix computation was born: Floyd's and Warshall's early-1960s shortest-path algorithms already computed exactly the tropical matrix powers described above, decades before anyone called this algebra "tropical."

Tropical geometry sits at a genuine crossroads. Every tropical polynomial $p$ carries a Newton polytope $\mathrm{conv}(S) \subseteq \mathbb{R}^n$ , and the coefficients $c_\alpha$ induce a regular subdivision of that polytope (lift each $\alpha$ to height $c_\alpha$ and project the lower faces back down) — exactly the polyhedral machinery of convex and discrete geometry. Dually, the tropical variety $\mathrm{trop}(X)$ of a classical algebraic variety $X$ encodes genuine algebraic-geometric data of $X$ (its dimension, degree, and much of its intersection theory) inside a purely combinatorial object, which is why so many computations in algebraic geometry can be carried out tropically. And because every tropical polynomial is a minimum of affine-linear functions, evaluating one is itself a small linear program: tropical geometry and convex/linear optimization share the same underlying $(\min, +)$ -linear structure, and tropical methods now feed directly into linear and convex programming algorithms.

Honesty first: this is a generic tetrahedron, one of the library's stock 3D polytopes — it is not the Newton polytope of any specific tropical polynomial from this page. Real Newton polytopes (of the tropical line, a plane conic, or the graph example above) are typically much simpler flat polygons in

\mathbb{R}^2

\mathbb{R}^3

, not this shape. What this widget honestly illustrates is only the general idea of a 3D convex polytope with flat faces that can be pulled apart (exploded) — the same kind of object (Newton polytopes, subdivided by height functions) that underlies the bridge to convex and discrete geometry described above.

AdvancedA hundred years in the making: from automata theory to algebraic geometry

Historical note

The min-plus semiring was discovered independently at least twice. In operations research, Robert Floyd (June $1962$ , Algorithm 97: Shortest Path) and Stephen Warshall (January $1962$ , A Theorem on Boolean Matrices) published the shortest-path and transitive-closure algorithms now jointly known as Floyd–Warshall, building on a $1959$ paper of Bernard Roy — all, in retrospect, computations of $(\min,+)$ matrix powers, though none of these authors used that language. In theoretical computer science, Imre Simon's $1978$ paper Limited subsets of a free monoid introduced min-plus weighted automata to decide the finite power property of regular languages, the direct ancestor of the semiring later named after him. These two traditions ran in parallel for decades. Only in the 2000s did mathematicians including Grigory Mikhalkin, Bernd Sturmfels, Diane Maclagan, and David Speyer assemble the min-plus semiring, Newton polytopes, and non-archimedean valuations into the geometric theory now called tropical geometry — with Mikhalkin's *Enumerative tropical algebraic geometry in $\mathbb{R}^2$ * ( $2005$ ) as a landmark: it proved a correspondence theorem showing that classical counts of complex algebraic curves through generic points can be computed exactly by counting suitably weighted tropical curves, turning what had been a combinatorial curiosity into a genuine tool for enumerative algebraic geometry.

ResearchWhere tropical geometry stands now

Research frontier as of 2026

As of 2026, tropical geometry remains an active crossroads rather than a closed chapter. Three directions stand out. (1) Enumerative algebraic geometry. Mikhalkin's $2005$ correspondence theorem was the first of many: tropical curve-counting techniques continue to be extended to higher-genus and higher-dimensional enumerative problems, feeding into the computation of Gromov–Witten invariants and other curve-counting theories where classical methods are hard to compute directly. (2) Computational biology. Because an evolutionary tree with branch lengths is itself a kind of tropical/piecewise-linear object, spaces of phylogenetic trees (such as Billera–Holmes–Vogtmann tree space) embed naturally into tropical linear spaces and tropical Grassmannians; this lets researchers replace some intractable combinatorial searches in maximum-likelihood and Bayesian phylogenetic inference with tropical convexity and tropical median computations. (3) Non-archimedean and Berkovich geometry. Tropicalizations are now understood as combinatorial skeletons sitting inside the richer analytic Berkovich spaces associated to varieties over non-archimedean fields; ongoing work (including differential forms, potential theory, and Monge–Ampère equations on Berkovich spaces, and the top-weight cohomology of moduli spaces of curves via tropical/Berkovich dual complexes) continues to tighten the dictionary between tropical combinatorics and non-archimedean analytic geometry. Open problems include extending correspondence theorems to broader classes of varieties and better understanding which tropical varieties actually arise as tropicalizations of classical ones (the realizability question).

In the min-plus convention used on this page, what is $2 \oplus 5$ ?

The tropical curve (corner locus) defined by a tropical polynomial $p$ is the set of points where...

The balancing condition at a vertex of a tropical curve, $\sum_j w_j v_j = 0$ , is important because...

The Floyd–Warshall shortest-path algorithm is a direct historical ancestor of tropical geometry because it essentially computes...

References

Diane Maclagan, Bernd Sturmfels (2015). Introduction to Tropical Geometry
Grigory Mikhalkin (2005). Enumerative tropical algebraic geometry in R^2 · arXiv:math/0312530
Imre Simon (1978). Limited subsets of a free monoid · DOI:10.1109/SFCS.1978.21