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Tropical geometry

Tropical geometry is a branch of that studies algebraic varieties over fields equipped with a non-Archimedean valuation through a process known as tropicalization, yielding piecewise-linear analogues in the form of polyhedral complexes. This framework replaces the usual of real numbers with the minimum (or maximum) operation and ordinary with multiplication, operating within the tropical , typically \mathbb{R} \cup \{\infty\} where x \oplus y = \min(x, y) and x \otimes y = x + y. As a combinatorial shadow of classical algebraic geometry, it simplifies complex problems by reducing them to polyhedral geometry and graph theory, often providing explicit counts and structures that are difficult to obtain directly. The field originated in the 1970s from Imre Simon's development of min-plus algebra for optimization problems, with the name "tropical" coined around 2000 by French computer scientists at the in honor of Simon's Brazilian heritage. Early foundational ideas appeared in George Bergman's 1971 work on logarithmic limit sets of algebraic varieties and in Robert Bieri and Burt Strebel's 1980 studies of valuations on groups. The modern formulation emerged in the early 1990s through Mikhael Kapranov's unpublished manuscript on tropical hypersurfaces and amoebas, gaining prominence in the 2000s with Mikhalkin's enumerative applications and Viro's patchworking techniques for real curves. By the , it had become a mature area, as detailed in comprehensive texts like Maclagan and Sturmfels' introduction. Central concepts in tropical geometry include tropical polynomials, which are finite maxima (or minima) of monomials and define (or ) piecewise-linear functions, and their zero loci, known as tropical hypersurfaces, which form balanced polyhedral complexes in . A tropical is the support of such a complex arising from the tropicalization of an in a over a valued , capturing the combinatorial type of the original while being invariant under the valuation. Key structures also encompass amoebas—the images of varieties under logarithmic maps—and matroids, which provide a foundation linking tropical geometry to . Tropical geometry has notable applications across mathematics and beyond, including the computation of Gromov–Witten invariants for enumerative curve counting via piecewise-linear correspondences, as pioneered by Mikhalkin in 2005. It aids in solving implicitization problems in computer-aided and analyzing phylogenetic trees in through tropical convex hulls. Additionally, its connections to optimization yield efficient algorithms for shortest paths and network flows, while in , it models hidden variables and contingency tables. These applications underscore its role as a bridge between continuous geometry, discrete mathematics, and applied sciences.

Historical Development

Origins in Algebraic Geometry

The study of amoebas of algebraic varieties originated in the context of non-Archimedean valuations applied to complex , providing a bridge to tropical structures through degeneration processes. The of a subvariety V \subset (\mathbb{C}^*)^n is defined as the image under the logarithmic map \mathrm{Log}: (\mathbb{C}^*)^n \to \mathbb{R}^n, where \mathrm{Log}(z) = (\log |z_1|, \dots, \log |z_n|), which encodes the non-Archimedean valuation on \mathbb{C}. This construction highlights the asymptotic behavior of varieties as points approach the boundary of the , revealing polyhedral "skeletons" and tentacles that foreshadow tropical varieties. Formally introduced by Gelfand, Kapranov, and Zelevinsky in 1994, amoebas were motivated by problems in discriminant theory and offered new insights into the and of algebraic sets, with early explorations of related logarithmic images appearing in valuation-theoretic contexts from the 1970s and 1980s. A key precursor to tropical methods arose from Mikhalkin's development of patchworking for real algebraic curves around 1997, which constructed real curves by piecing together local pieces from a grid using sign conditions and complex topology. This technique, building on Viro's earlier patchworking for hypersurfaces, emphasized the discrete combinatorial assembly of real varieties and demonstrated how real enumerative invariants could be computed via piecewise linear approximations, anticipating the tropical limit of amoebas as the scaling parameter tends to zero. Mikhalkin's approach revealed that the topology of real curves in toric surfaces corresponds to multiplicities on dual graphs, providing a combinatorial framework that directly influenced the emergence of tropical curve counting. Berkovich spaces, introduced by Vladimir Berkovich in 1990, formalized non-Archimedean over complete valued fields, offering a topological framework to realize tropical limits as degenerations of families of varieties. These spaces analytify algebraic varieties over non-Archimedean fields, and their "tropical skeletons"—retractions onto polyhedral complexes—capture the combinatorial essence of the variety in the limit as the valuation becomes trivial. This construction justifies tropicalization as a faithful degeneration, where the Berkovich analytification retracts onto a metric graph or polyhedral complex homeomorphic to the tropical variety, enabling rigorous study of limits in higher dimensions beyond amoebas. An illustrative example of this interplay is the relation between the Newton and tropicalization through initial ideals: for a ideal I \subset k[x_1^{\pm 1}, \dots, x_n^{\pm 1}] over a valued , the tropical \mathrm{Trop}(V(I)) consists of those weights w \in \mathbb{R}^n such that the initial ideal \mathrm{in}_w(I), generated by the lowest-valuation terms with respect to w, is proper (not the unit ideal). The faces of the Newton of generators of I correspond to these initial forms, and the tropical emerges as the codimension-one skeleton where \mathrm{in}_w(f) is not a for a defining f, thus encoding the polyhedral structure directly from the valuation-induced degeneration.

Key Milestones and Contributors

Tropical geometry emerged as a distinct field in the early , with Grigory Mikhalkin playing a pivotal role in its introduction through his work on enumerative invariants. In 2003, Mikhalkin developed the foundational framework for tropical geometry to address enumerative problems in , particularly the counting of curves in toric surfaces. His approach replaced complex algebraic structures with piecewise-linear objects in , enabling combinatorial methods to compute Gromov-Witten invariants via lattice paths in Newton polygons. This seminal contribution, published in 2005, established tropical curves as stable limits of complex curves and provided explicit formulas for curve enumeration of arbitrary . Parallel to Mikhalkin's geometric innovations, Bernd Sturmfels and collaborators advanced the algebraic underpinnings of the field around 2003–2005, focusing on tropical linear algebra. Sturmfels, along with David Speyer, introduced concepts such as tropical matrix rank and linear systems over the tropical semiring, adapting classical linear algebra to min-plus arithmetic. Their work demonstrated how tropical solutions to linear equations correspond to stable intersections in algebraic geometry, laying the groundwork for broader applications in optimization and statistics. A key outcome was the exploration of tropical eigenvectors and eigenvalues, which revealed combinatorial structures underlying classical problems. Significant milestones in the mid-2000s included the formalization of core structural properties for tropical objects. In 2005, Mikhalkin defined the balancing for tropical curves, a local compatibility requirement at vertices that ensures the polyhedral complex mimics the of its classical counterpart. This , essential for embedding tropical geometry within algebraic frameworks, facilitated proofs of enumerative correspondences and extended patchworking techniques to higher genera. Concurrently, in 2003–2004, and Sturmfels introduced the tropical , parametrizing tropical linear spaces as polyhedral fans derived from Plücker relations. Their 2004 publication showed that the tropical G_{2,n} coincides with the space of phylogenetic trees, bridging tropical geometry with and , while higher-dimensional cases revealed intricate simplicial complexes. The field's maturation was marked by comprehensive texts synthesizing these developments. In 2015, Diane Maclagan and Bernd Sturmfels published Introduction to Tropical Geometry, a foundational graduate-level book that unified polyhedral and algebraic perspectives, covering valuations, semirings, and varieties with exercises and recent results. Maclagan contributed significantly to refining tropical schemes and ideals, developing moduli spaces for tropical curves and hypersurfaces that incorporate multiplicities and stability conditions. Key contributors include Grigory Mikhalkin, whose enumerative theorems and curve theory provided the geometric impetus; Bernd Sturmfels, who drove algebraic and computational advancements, including linear spaces and statistical models; and Diane Maclagan, whose work on tropical and schemes enhanced the field's rigor and applicability. These figures, through seminal papers and collaborations, transformed tropical geometry from a niche tool into a vibrant interdisciplinary domain.

Algebraic Foundations

Valuations and Valued Fields

A valuation on a K is a function \mathrm{val}: K \to \Gamma \cup \{\infty\}, where \Gamma is an ordered , satisfying \mathrm{val}(0) = \infty, \mathrm{val}(ab) = \mathrm{val}(a) + \mathrm{val}(b) for all a, b \in K, and \mathrm{val}(a + b) \geq \min\{\mathrm{val}(a), \mathrm{val}(b)\} for all a, b \in K, with equality holding when \mathrm{val}(a) \neq \mathrm{val}(b). This multiplicative property and the ultrametric inequality ensure that the valuation captures a notion of "size" or "order of vanishing" in the . Examples include the [p-adic valuation](/page/P-adic_valuation) on the rational numbers \mathbb{Q}, for a prime p, defined by \mathrm{val}_p(p^k \cdot \frac{a}{b}) = k where a, b \in \mathbb{Z} are not divisible by p and k \in \mathbb{Z}. Another is the trivial valuation, where \mathrm{val}(0) = \infty and \mathrm{val}(a) = 0 for all a \in K \setminus \{0\}. These illustrate both nontrivial archimedean-like behavior in completions and the degenerate case where no elements are distinguished by size. A valued field is a pair (K, \mathrm{val}) consisting of a K and a valuation \mathrm{val} on K. The value group of (K, \mathrm{val}) is the image \mathrm{val}(K \setminus \{0\}) \subseteq \Gamma, an ordered of \Gamma. The is the of the valuation ring \{a \in K \mid \mathrm{val}(a) \geq 0\} by its maximal ideal \{a \in K \mid \mathrm{val}(a) > 0\}. These structures decompose the valued field into a "skeleton" provided by the value group and a "body" given by the , facilitating analysis in algebraic geometry. Non-Archimedean valuations, characterized by the ultrametric inequality \mathrm{val}(a + b) \geq \min\{\mathrm{val}(a), \mathrm{val}(b)\}, play a crucial role in studying degenerations of families of algebraic varieties by encoding limits where certain terms dominate others. In this context, they model multi-parameter degenerations through higher-rank extensions, linking to tropical structures via tangent cones and refined tropicalizations. For a polynomial f = \sum_i a_i x^i \in K, the tropicalization is the tropical polynomial \mathrm{trop}(f)(y) = \min_i \{ \mathrm{val}(a_i) + i y \} for y \in \mathbb{R}, capturing the initial degeneration behavior under the valuation. This min-plus operation underpins the tropical semiring derived from valued fields.

Tropical Semiring

The tropical semiring, also known as the min-plus semiring, provides the foundational for tropical geometry by reinterpreting classical and in a way that aligns with piecewise-linear geometry. It is defined on the set \mathbb{T} = \mathbb{R} \cup \{\infty\}, where the tropical \oplus is given by a \oplus b = \min(a, b) and the tropical \otimes is given by a \otimes b = a + b. This structure arises naturally from the valuation on fields, embedding classical algebraic objects into a tropical setting via limits or degenerations. The tropical semiring is commutative under both operations, with \oplus and \otimes being associative: for all a, b, c \in \mathbb{T}, a \oplus (b \oplus c) = (a \oplus b) \oplus c and a \otimes (b \otimes c) = (a \otimes b) \otimes c. It satisfies the distributive law: a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c). The additive identity, or zero element, is \infty, since a \oplus \infty = \min(a, \infty) = a for all a \in \mathbb{T}, while the multiplicative identity, or unit element, is $0, as a \otimes 0 = a + 0 = a. The term "tropical" was coined in the 1990s by French mathematicians, including Jean-Éric Pin, in honor of Brazilian mathematician Imre Simon, building on earlier work in idempotent analysis and optimization. A distinctive property is the of addition: a \oplus a = \min(a, a) = a for all a \in \mathbb{T}, which contrasts with classical rings and leads to "flat" or piecewise-linear phenomena in tropical geometry. Additionally, \infty acts as an absorbing under : a \otimes \infty = a + \infty = \infty for all a \in \mathbb{T}. These features make the tropical semiring an idempotent , lacking additive inverses but sufficient for defining tropical polynomials and varieties. Generalizations extend the tropical semiring beyond scalars. For instance, it can be defined on \mathbb{R}^n \cup \{\infty\} by componentwise operations, enabling tropical vector spaces and modules. An isomorphic variant, the max-plus semiring, uses \mathbb{R} \cup \{-\infty\} with \oplus as \max(a, b) and \otimes as a + b, related by negation x \mapsto -x. Further extensions incorporate coefficients from valued fields, such as Puiseux series, to model tropicalizations of algebraic varieties over non-Archimedean fields. Tropical matrix operations illustrate linear algebra over the . For matrices A, B \in \mathbb{T}^{m \times n}, the product C = A \otimes B has entries c_{ij} = \bigoplus_{k=1}^n (a_{ik} \otimes b_{kj}) = \min_k (a_{ik} + b_{kj}), representing shortest paths in or optimal assignments in . The tropical rank of a matrix, such as the Barvinok rank, measures the dimension of its image under these operations, with examples like the cycle matrix C_6 having rank 4.

Core Concepts

Tropical Polynomials

In tropical geometry, a tropical polynomial in one variable over the tropical real numbers \mathbb{T} = \mathbb{R} \cup \{\infty\} is defined using the tropical semiring operations of minimum for addition and standard addition for multiplication. Specifically, for coefficients a_0, a_1, \dots, a_d \in \mathbb{R}, the polynomial is P(x) = \bigoplus_{i=0}^d a_i \otimes x^i = \min_{i=0}^d (a_i + i x), where terms with coefficient \infty are ignored. This expression arises from the tropicalization of classical polynomials via valuations on fields like \mathbb{C}\{t\}, where the coefficients a_i correspond to the valuations of the original coefficients. The graph of P(x) is a continuous, piecewise linear function with segments of integer slopes ranging from 0 to d, ordered increasingly from left to right, as higher-degree terms (steeper slopes) dominate for sufficiently negative x, while the constant term dominates for large positive x. The breakpoints, or bends, occur where the minimizing term changes, resulting in a whose pieces form the lower envelope of the affine lines y = a_i + i x. The of P is the largest i such that a_i is finite, and by the tropical analog of the , P(x) factors uniquely as a tropical product of d linear factors (x \oplus r_j), counting multiplicities. Tropical roots of P(x) are the values of x \in \mathbb{R} where the minimum is attained by at least two terms, corresponding precisely to the in the . At such a r, the multiplicity is the between the outgoing (to the right) and incoming (to the left) of the adjacent linear pieces; for instance, a change from 2 to 0 yields multiplicity 2, while a change from 2 to 1 followed by 1 to 0 each has multiplicity 1. The total number of , counted with multiplicity, equals the degree d. As an illustrative example, consider the quadratic tropical polynomial P(x) = 0 \oplus 1 \otimes x \oplus 3 \otimes x^2 = \min(0, 1 + x, 3 + 2x). The graph consists of three pieces: for x \leq -2, P(x) = 3 + 2x (slope 2); for -2 \leq x \leq -1, P(x) = 1 + x (slope 1); and for x \geq -1, P(x) = 0 (slope 0). The bends occur at the roots x = -2 (where $1 + x = 3 + 2x, multiplicity 1) and x = -1 (where $1 + x = 0, multiplicity 1), confirming the degree 2.

Tropical Hypersurfaces

In tropical geometry, a is defined as the zero set of a . Given a tropical polynomial P in n variables over the tropical (\mathbb{R} \cup \{\infty\}, \min, +), the tropical hypersurface V(P) is the set \{ x \in \mathbb{R}^n \mid the minimum value of P(x) is achieved by at least two monomials \}. This locus consists of points where the piecewise-linear function P(x) is not smooth, forming the "corners" of the graph. The tropical hypersurface V(P) is a balanced polyhedral of pure n-1, hence codimension 1 in \mathbb{R}^n. Its facets are the maximal cells, each corresponding to pairs of that achieve the minimum simultaneously, while the complementary regions in \mathbb{R}^n are where a single dominates. The is balanced, meaning that at each v of codimension n-1, the sum of the outgoing vectors along the adjacent edges, weighted by their respective multiplicities ( indices of the facets), equals the zero vector: \sum u_i m_i = 0, where u_i are the primitive vectors and m_i the multiplicities. This condition ensures the polyhedral structure mimics the balancing of normal fans in . The structure of V(P) is intimately tied to the Newton polytope \Delta(P), the of the exponent vectors of the monomials in P. The hypersurface induces a dual regular subdivision of \Delta(P), where the cells of V(P) are in order-reversing with the faces of this subdivision; specifically, unbounded rays in V(P) correspond to vertices on the of \Delta(P), and the subdivision arises from the lower under the weights given by the coefficients of P. This duality allows computational algorithms to construct V(P) by first computing the normal of \Delta(P) and intersecting it with the at height 1. A representative example is the tropical line in \mathbb{R}^2, defined by the polynomial P(x,y) = "x" \oplus "y" \oplus 0, where the coefficients are taken as 0 for simplicity (shifting the vertex to the ). Here, V(P) consists of three unbounded rays emanating from the : one along the negative x-axis (direction (-1,0)), one along the negative y-axis (direction (0,-1)), and one along the line x + y = 0 in the first (direction (1,1)). The balancing condition holds at the , as the primitive vectors satisfy (-1,0) + (0,-1) + (1,1) = (0,0), each with multiplicity 1. This structure subdivides the Newton triangle (with vertices at (0,0), (1,0), and (0,1)) into three smaller triangles, dual to the rays.

Tropical Varieties

Tropical varieties arise as the tropicalization of classical algebraic varieties over valued s. Consider a k with a non-archimedean valuation \val: k^\times \to \mathbb{R} \cup \{\infty\}, which is algebraically closed, such as the field of Puiseux series \mathbb{C}\{t\}. For an ideal I \subseteq k[x_1, \dots, x_n], the variety V(I) \subseteq k^n is considered within the (k^\times)^n, and the tropical variety is defined as \trop(I) = \overline{\val(V(I) \cap (k^\times)^n)} \subseteq \mathbb{R}^n, where the closure is in the on \mathbb{R}^n. This construction captures the "skeleton" or degenerate limit of the variety under the valuation map \val: (k^\times)^n \to \mathbb{R}^n, given by \val(a_1, \dots, a_n) = (\val(a_1), \dots, \val(a_n)). An equivalent algebraic characterization uses initial ideals with respect to weight vectors. For w \in \mathbb{R}^n, the initial part of a Laurent polynomial f \in k[x_1^{\pm 1}, \dots, x_n^{\pm 1}] is the \in_w(f) consisting of terms achieving the minimum w-weighted valuation \min_{c \mathbf{x}^\alpha \in \mathrm{supp}(f)} (\val(c) + \langle w, \alpha \rangle), and the initial ideal is \in_w(I) = (\in_w(f) \mid f \in I). Then, w \in \trop(I) if and only if \in_w(I) contains no pure , i.e., \in_w(I) \cap k[x_1^{\pm 1}, \dots, x_n^{\pm 1}] has no generators. Equivalently, considering a flat family I_t over t \in k with \val(t) \to -\infty, points in \trop(V(I)) correspond to those w where the initial ideals \in_{t w}(I_t) are monomial-free for generic small t. Tropical varieties satisfy the stable intersection property: for ideals I, J \subseteq k[x_1^{\pm 1}, \dots, x_n^{\pm 1}], \trop(I \cap J) = \trop(I) \cap \trop(J). This allows tropical varieties to be realized as finite intersections of tropical hypersurfaces, preserving the combinatorial structure of classical intersections under generic conditions. Multiplicities on the codimension-one facets of \trop(I) are defined combinatorially: for w in the relative interior of a facet, the multiplicity is \sum_{P \in \mathrm{Ass}(\in_w(I))} \mult(P, \in_w(I)), where \mathrm{Ass} denotes the associated primes and \mult is the multiplicity of the prime ideal. In enumerative contexts, these multiplicities can also be computed via mixed volumes of the polytopes of generators of I, providing a link to classical . A example is the tropicalization of s defined by principal ideals in k[x, y]. For a such as p(x,y) = a x^2 \oplus b x y \oplus c y^2 \oplus d x \oplus e y \oplus f over the tropical , under suitable valuation conditions on the coefficients (e.g., distinct valuations), the resulting tropical curve is a in \mathbb{R}^2 with four vertices, three bounded edges, and six unbounded rays emanating in the directions (1,0), (0,1), and (-1,-1). More generally, the tropical of d from V(f) where f is a -d intersects a generic tropical line in exactly d points, counting multiplicities, illustrating the stable intersection property in dimension two.

Geometric Structures

Polyhedral Complexes

In tropical geometry, polyhedral complexes provide the combinatorial framework for realizing tropical varieties as geometric objects. A tropical variety is defined as the support of a pure-dimensional polyhedral complex equipped with a rational affine structure, meaning that each maximal cell is an affine subspace over the rationals, and the complex is invariant under the tropical action. This structure ensures that the complex is a balanced weighted polyhedral complex of constant dimension, where the weights are positive integers assigned to each facet. The defining feature of such complexes is the balancing condition, which generalizes the classical Plücker relations in a combinatorial setting. Specifically, for a facet \sigma of codimension one in the complex \Sigma, the sum of the outgoing primitive lattice vectors u_i from the adjacent maximal cells, weighted by their multiplicities m_i, satisfies \sum m_i u_i = 0 in the quotient lattice N_\sigma = N / N_\sigma, where N is the lattice containing the complex. This condition holds at every codimension-one face and ensures a form of "zero-tension" analogous to equilibrium in physical systems. Dimension in the complex is defined as the dimension of its maximal cells, while codimension refers to the relative dimension of faces within the ambient space, with the complex being pure-dimensional to maintain uniformity. Examples of such polyhedral complexes include fans, which arise from the Bergman fan of a and satisfy the balancing condition intrinsically due to their combinatorial nature. Tropical linear spaces, such as the tropical , form another class, where the complex is a fan in the tropical and the balancing reflects the matroidal structure of the underlying linear space. These examples illustrate how abstract combinatorial objects embed into the tropical framework as balanced complexes. A key result in tropical geometry is the uniqueness theorem: every tropical variety, obtained as the tropicalization of an algebraic variety over a valued field, uniquely determines a polyhedral complex that is balanced and satisfies the rational affine structure condition. This equivalence bridges the algebraic origins of tropical varieties with their polyhedral realizations, allowing combinatorial study without reference to the underlying classical geometry.

Tropical Curves

Tropical curves are one-dimensional tropical varieties, realized as metric graphs embedded in Euclidean space, consisting of vertices connected by edges of finite or infinite length, where the latter are unbounded rays extending to infinity. These graphs satisfy a balancing condition at every vertex of valence at least three: the sum of the primitive integer direction vectors of the incident edges, weighted by their positive integer multiplicities, equals zero in the ambient space. This condition ensures the graph's stability and mirrors the divisor properties of algebraic curves in non-Archimedean fields. Within the broader framework of polyhedral complexes, tropical curves provide a combinatorial model for degenerations of algebraic curves. The of a tropical curve is defined as the number of independent cycles in its underlying , equivalent to the first of the viewed as a . For a connected with v vertices and e edges (counting unbounded edges as contributing one endpoint at ), the g satisfies g = e - v + 1. This topological invariant captures the curve's complexity, analogous to the of a classical , and remains preserved under stable equivalence relations in moduli spaces. Plane tropical curves are embeddings of these metric graphs into \mathbb{R}^2, where edges have rational slopes and carry positive integer multiplicities reflecting the lattice length in the dual subdivision of the . These multiplicities determine numbers and ensure compatibility with the balancing condition, allowing the curve to model limits of families of algebraic plane curves. Such embeddings are proper, meaning unbounded edges extend in fixed directions corresponding to the or their negatives. A key enumerative result is Mikhalkin's correspondence theorem, which equates the number of plane tropical curves of given d and g passing through $3d + g - 1 points in \mathbb{R}^2—counted with multiplicity—to the number of complex algebraic curves of the same degree and genus in the corresponding toric surface passing through the analogous configuration. This bijection, established via lattice path counts in the \Delta_d, translates algebraic invariants like Gromov-Witten numbers into purely combinatorial data from tropical curves. An illustrative example is the tropical elliptic curve, a genus-one plane tropical curve realized as a single cycle equipped with three unbounded legs emanating from vertices on the cycle, each of weight one and directed along the negative coordinate axes. This structure arises as the tropicalization of a classical elliptic curve in \mathbb{CP}^2, with the cycle length encoding the j-invariant in the tropical setting.

Intersection Theory

In tropical geometry, intersection theory provides tools to compute the number and multiplicities of intersection points between tropical varieties, analogous to classical but using polyhedral structures and balancing conditions. For hypersurfaces defined by tropical polynomials, intersections are analyzed through stable intersections, where varieties are perturbed slightly to ensure transverse crossings, preserving the total intersection number. This framework extends to higher dimensions via polyhedral complexes, enabling the study of multiplicities via lattice data. The tropical Bézout theorem states that two plane tropical curves of degrees d and d' intersect in exactly d \cdot d' points, counting multiplicities, under intersection conditions. This holds even if the curves share components, by considering limits of nearby positions. The theorem mirrors the classical Bézout theorem and is fundamental for enumerative problems in tropical geometry. Multiplicities at points are defined locally for intersections. When two edges of tropical curves, with lattice directions u and u' and weights w and w', intersect transversely at a point p, the multiplicity is w \cdot w' \cdot |\det(u, u')|, where \det(u, u') measures the lattice of the parallelogram spanned by the directions. This relies on the balancing at vertices and ensures invariance under perturbations. For non-transverse cases, such as shared edges, the multiplicity along a common edge segment is given by the lattice length of that segment multiplied by the product of the edge weights. As an example, consider the of two tropical curves sharing a finite common edge of length \ell. The multiplicity contributed by this shared segment is \ell, assuming unit weights on the edges; this accounts for the "unstable" overlap in the total Bézout count when resolved. On tropical curves, is developed through theory. A on a tropical curve \Gamma is a formal \mathbb{Z}- of points on \Gamma, with effective divisors having non-negative coefficients. Principal divisors arise from rational functions on \Gamma, which are linear with slopes, and is defined modulo these principal divisors. The intersection of a divisor with a subcurve or point is computed via orders of vanishing, leading to a tropical \mathrm{Pic}(\Gamma) that classifies line bundles. Chow rings for tropical curves extend this to codimension-1 cycles, generated by divisors modulo rational equivalence (linear equivalence in this case), with ring structure given by intersection products on the curve. For a smooth tropical curve of genus g, the Chow ring A^*(\Gamma) is isomorphic to \mathbb{Z} \oplus \mathrm{Pic}^0(\Gamma) \oplus \mathbb{Z}/g\mathbb{Z}, where \mathrm{Pic}^0(\Gamma) parameterizes degree-zero divisors, facilitating computations of intersection numbers via Riemann-Roch analogs. For higher-codimension intersections, such as the intersection of multiple hypersurfaces, the Cayley trick embeds the problem into a mixed subdivision of a Cayley configuration in one higher dimension. This reduces the computation to a triangulation of the product of simplices, where the tropical variety corresponds to a coherent mixed subdivision, allowing multiplicities to be read from volumes of cells in the subdivision. The trick preserves the balanced fan structure and is essential for non-complete intersection cases.

Applications and Connections

Enumerative and Combinatorial Geometry

Tropical geometry provides powerful tools for solving enumerative problems in , particularly through correspondences between classical invariants and counts of tropical objects. A key result is Mikhalkin's correspondence theorem, which equates the Gromov-Witten invariants of toric surfaces—counting complex curves of given degree and genus passing through generic points—with weighted enumerations of tropical curves satisfying analogous incidence conditions. This theorem, building on ideas from Kontsevich's work on stable maps, allows computation of these invariants via combinatorial means, such as multiplicities determined by counts in the dual subdivision of the . For instance, the number of plane rational curves of degree d through $3d-1 points is given by a tropical count that matches the classical value, like 1 for d=1, 1 for d=2, and 12 for d=3. In combinatorial geometry, tropical Grassmannians connect matroid theory to tropical varieties, realizing matroid polytopes as Bergman fans of these spaces. The tropical Grassmannian \mathrm{Trop}(Gr(k,n)), obtained as the tropicalization of the of k-planes in n-space, parametrizes tropical linear spaces arising from classical ones and is defined by tropical Plücker relations. Matroids, abstracting linear dependence, have their base polytopes as the images under the moment map of orbits in Grassmannians; tropicalizing these yields polyhedral fans whose cones correspond to matroid flats, enabling realizations of matroid polytopes as tropical varieties over fields with valuation. This framework has facilitated studies of matroid orientations and valuations on subdivisions, linking to Hopf algebra structures in . The Dressian \mathrm{Dr}(k,n), introduced by Dress and extended tropically, parametrizes all tropical linear spaces via tropical Plücker vectors satisfying the three-term relations, forming a fan in the space of Plücker coordinates. In contrast, the tropical Grassmannian \mathrm{Trop}(Gr(k,n)) only includes those realizable over a field, making the Dressian a larger space that encompasses non-realizable matroids; for example, \mathrm{Dr}(3,6) has more rays than \mathrm{Trop}(Gr(3,6)), reflecting non-Pappus configurations. Recent results show equality in the positive orthant, where the positive Dressian coincides with the positive tropical Grassmannian, aiding positivity studies in total positivity and cluster algebras. Tropical methods have yielded new complexity bounds for in the 2020s, analyzing interior-point and algorithms via tropical geometry. By tropicalizing the central path of barrier methods, researchers derived explicit upper bounds on iteration complexity, such as O(\sqrt{n} \log(1/\epsilon)) for self-concordant barriers, matching classical rates but with tropical proofs revealing geometric obstructions like non-smoothness in high dimensions. These approaches also apply to mean-payoff games and parametric optimization, where tropical duals provide combinatorial certificates for optimality. A representative enumerative example is the count of bitangents to plane quartic curves, classically 28 in the case. Tropically, a smooth plane quartic curve admits exactly seven bitangent classes, each a combinatorial type of tropical line tangent at two points; weighted counts of these classes, considering multiplicities from balancing conditions, recover the classical invariant via Mikhalkin's correspondence extended to higher tangency. For real quartics, tropical methods refine this to signed counts, yielding Welschinger-type invariants that track real bitangents, with computational tools enumerating the 41 possible shapes up to symmetry.

Biological and Physical Applications

Tropical geometry has found significant applications in , where the space of phylogenetic trees can be modeled as a tropical linear space within the tropical . Specifically, the tropical \mathrm{Gr}(2,n) parametrizes the space of all phylogenetic trees on n labeled leaves, establishing a connection between tree metrics and tropical varieties that dates back to foundational work in the mid-2000s. In this framework, dissimilarity maps—representations of pairwise distances between taxa derived from evolutionary trees—lie on the tropical , allowing for the of tree topologies and evolutionary relationships. Tree metrics themselves arise as points in tropical convex hulls, enabling the study of convex combinations of trees that correspond to statistical mixtures in phylogenetic inference, such as in models or ancestral . A key example is the use of tropical distance measures for evolutionary trees, which quantify differences between phylogenies via the tropical metric on dissimilarity maps. This metric, defined in the max-plus , captures the Robinson-Foulds distance or path differences in tree space, facilitating tasks like tree and clustering in large-scale genomic . For instance, tropical balls in the space of phylogenetic provide neighborhoods for nearest-neighbor searches, improving computational efficiency in inferring evolutionary histories from dissimilarity vectors. These tools have been applied since the mid-2000s to handle the in tree space, offering a polyhedral structure that supports algorithms for reconstructing phylogenies from molecular sequence . In the context of mirror symmetry, tropical geometry provides a combinatorial for degenerations of Calabi-Yau manifolds, bridging algebraic and geometries. Tropical curves serve as skeletons that encode the limiting behavior of complex curves in toric Calabi-Yau degenerations, allowing the construction of mirror partners via real affine manifolds with singularities—often termed tropical manifolds. This approach, part of the Gross-Siebert program, resolves central problems in mirror symmetry by producing explicit degenerations where tropical data dictates the and period integrals of the mirror family. For example, monomial ideals in toric Fano varieties yield tropical mirrors that match constructions, confirming predictions from in specific cases like quintic hypersurfaces. Applications in physics, particularly in the 2010s, leverage tropicalizations of to geometrize amplitudes in quantum field . The , introduced as a positive geometry in the , encodes tree-level amplitudes in planar \mathcal{N}=4 Yang-Mills , with its combinatorial structure arising from the positive tropical . Tropical methods reveal the factorization properties and singularities of these amplitudes, providing a framework where volumes of tropical polytopes directly yield integrands for loop amplitudes. This tropical perspective has extended to broader contexts, such as algebraic singularities in symbol alphabets, linking and tropical to finite alphabets for higher-point amplitudes. Connections in the 2010s, such as Huh's 2012 work using the Bergman fan and the 2015 proof of Rota's by Adiprasito, Huh, and Katz using , highlight tropical geometry's role in proving log-concavity properties of s, with implications for and . The Bergman fan of a , a canonical tropical linear space, underlies the log-concavity of the coefficients in the 's characteristic , as established through algebro-geometric methods that tropicalize linear realizations. The full proof of Rota's was given in 2015 by Adiprasito, Huh, and Katz, showing that the Whitney numbers alternate in sign and form a log-concave sequence, with equality conditions tied to decompositions. Ongoing work explores ultra-log-concavity and applications to mixing times in Markov chains on polytopes.

Recent Developments

In recent years, computational tropical geometry has seen significant advancements through the development of efficient algorithms for constructing tropical varieties and solving optimization problems. A key initiative, funded by the UK Research and Innovation (UKRI), focused on leveraging tropical methods to address computational challenges in applied sciences, including improvements in homotopy continuation for polynomial systems and the use of saturating Gröbner bases to find positive real solutions efficiently. This project, extended through 2023, applied these algorithms to areas such as chemical reaction networks, phylogenetics, and financial liquidity allocation, demonstrating tropical geometry's potential for scalable optimization in industrial settings. Research on tropical abelian varieties and their Jacobians has gained prominence, particularly in exploring moduli spaces and degeneration techniques in . At the Women and Mathematics (WAM) program in 2025, Melody Chan's lectures introduced tropical curves alongside tropical abelian varieties, emphasizing their combinatorial structures and connections to classical abelian varieties via limiting processes. Concurrent work has advanced tropical Abel-Jacobi theory, defining functorial maps and introducing tropical varieties as analogs to classical s for curves. These developments extend by providing tropical frameworks for understanding Jacobian decompositions in higher genera. Emerging connections between tropical geometry, log-concavity, and theory have been highlighted in interdisciplinary workshops, revealing deep links between combinatorial independence structures and geometric properties. The 2025 WAM program at the Institute for Advanced Study featured Josephine Yu's lectures on matroids and log-concave polynomials, illustrating how tropical methods model the real and combinatorial geometry underlying log-concavity, with applications to mixing times in random walks on matroids. This synthesis positions tropical geometry as a bridge for proving log-concavity in multivariate settings relevant to optimization and probability. New directions in tropical geometry include explorations of tropical Hodge theory and non-Archimedean motives, which integrate combinatorial tools with analytic and arithmetic perspectives. The Oberwolfach workshop on "Tropical Geometry: New Directions" discussed these areas as part of broader advancements, such as refined enumerative invariants and tropical , fostering connections to non-Archimedean for studying motives in degeneration limits. As an illustrative example, extensions of in the tropical setting have been examined to count intersections on tropical hypersurfaces with greater precision, accommodating multiplicities and higher-dimensional cases. A 2025 lecture at by Ralph Morrison highlighted these extensions, showing how tropical analogs preserve classical intersection multiplicities while enabling combinatorial proofs for enumerative problems beyond plane curves.