9.1 Tropicalization of algebraic varieties

Tropicalization transforms algebraic varieties into tropical varieties, revealing their combinatorial structure. This process uses non-archimedean valuations to convert algebraic equations into tropical ones, preserving key features like dimension and intersection properties.

Tropical varieties are piecewise-linear objects that encode information about algebraic varieties. They're defined as corner loci of tropical polynomials and have connections to combinatorics, optimization, and other areas of math. Understanding tropicalization is crucial for studying algebraic geometry through a combinatorial lens.

Algebraic varieties

• Algebraic varieties are geometric objects defined by polynomial equations and a fundamental object of study in algebraic geometry
• They can be classified into affine, projective, and quasi-projective varieties based on the type of ambient space they are embedded in
• Understanding the structure and properties of algebraic varieties is crucial for studying their tropical counterparts

Affine varieties

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• Defined as the zero locus of a set of polynomials in affine space $\mathbb{A}^n$
• Can be represented as the vanishing set of an ideal in a polynomial ring $k[x_1, \ldots, x_n]$
• Examples include curves (parabola $y = x^2$), surfaces (sphere $x^2 + y^2 + z^2 = 1$), and higher-dimensional objects

Projective varieties

• Defined as the zero locus of a set of homogeneous polynomials in projective space $\mathbb{P}^n$
• Described by homogeneous ideals in the homogeneous coordinate ring $k[x_0, \ldots, x_n]$
• Projective varieties are compact and have nice intersection properties (Bézout's theorem)
• Examples include projective curves (elliptic curves), projective surfaces (K3 surfaces), and Calabi-Yau manifolds

Quasi-projective varieties

• Defined as open subsets of projective varieties, obtained by removing a closed subvariety from a projective variety
• Can be represented as the complement of the zero locus of homogeneous polynomials in projective space
• Quasi-projective varieties provide a more general setting for studying algebraic geometry
• Examples include affine varieties (as open subsets of projective varieties) and complements of hypersurfaces in projective space

Tropicalization process

• Tropicalization is a process that associates a tropical variety to an algebraic variety, revealing combinatorial and geometric information
• It involves working over valued fields and using non-archimedean valuations to transform algebraic equations into tropical equations
• The tropicalization process preserves important features of the original algebraic variety, such as dimension and intersection properties

Puiseux series

• Puiseux series are formal power series with fractional exponents, used to study algebraic curves near a singularity
• They form an algebraically closed field extension of the field of Laurent series
• In the context of tropicalization, Puiseux series are used as the coefficient field for the valued field over which the algebraic variety is defined

Valued fields

• A valued field is a field $K$ equipped with a valuation map $v: K \rightarrow \mathbb{R} \cup \{\infty\}$ satisfying certain properties
• The valuation map measures the "size" or "order of magnitude" of elements in the field
• Examples of valued fields include the field of Puiseux series with the valuation given by the lowest exponent and the field of $p$-adic numbers with the $p$-adic valuation

Non-archimedean valuation

• A non-archimedean valuation is a valuation $v$ on a field $K$ satisfying the strong triangle inequality: $v(x + y) \leq \max\{v(x), v(y)\}$ for all $x, y \in K$
• Non-archimedean valuations are crucial in tropical geometry as they allow for the definition of tropical operations and the study of tropical limits
• The use of non-archimedean valuations distinguishes tropical geometry from classical algebraic geometry

Coordinate-wise valuation

• Given a valued field $(K, v)$ and a vector space $K^n$, the coordinate-wise valuation is defined as $v(x_1, \ldots, x_n) = (\min\{v(x_1), \ldots, v(x_n)\})$
• The coordinate-wise valuation extends the valuation on the field to the vector space, allowing for the tropicalization of algebraic sets
• The minimum in the definition of the coordinate-wise valuation is taken with respect to the total order on the value group of the valuation

Tropical varieties

• Tropical varieties are the central objects of study in tropical geometry, obtained as the tropicalization of algebraic varieties
• They are defined as the corner locus of tropical polynomials and exhibit a piecewise-linear structure
• Tropical varieties encode important information about the original algebraic varieties and have connections to combinatorics, optimization, and other areas of mathematics

Definition of tropical varieties

• A tropical polynomial is a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ given by $f(x) = \max\{a_1 + v_1 \cdot x, \ldots, a_k + v_k \cdot x\}$, where $a_i \in \mathbb{R}$ and $v_i \in \mathbb{Z}^n$
• The tropical hypersurface defined by a tropical polynomial $f$ is the set of points $x \in \mathbb{R}^n$ where the maximum in the definition of $f$ is attained at least twice
• A tropical variety is the intersection of finitely many tropical hypersurfaces

Tropical hypersurfaces

• Tropical hypersurfaces are the building blocks of tropical varieties, defined as the corner locus of a single tropical polynomial
• They have a polyhedral structure and can be represented as the dual of the Newton polytope of the tropical polynomial
• Examples include tropical lines ($\max\{x, y, 0\}$), tropical quadrics ($\max\{x + y, x, y, 0\}$), and higher-dimensional tropical hypersurfaces

Tropical curves

• Tropical curves are tropical varieties of dimension one, obtained as the tropicalization of algebraic curves
• They are represented as balanced weighted graphs, where the weights satisfy certain conditions at each vertex
• Tropical curves have been used to study enumerative geometry problems, such as counting curves satisfying certain incidence conditions

Tropical linear spaces

• Tropical linear spaces are tropical varieties that are tropicalizations of classical linear spaces
• They can be defined as the tropical vanishing locus of linear forms or as the intersection of tropical hyperplanes
• Tropical linear spaces have a rich combinatorial structure and are related to matroid theory and tropical Grassmannians

Tropicalization of algebraic sets

• The tropicalization of an algebraic set is a tropical variety that captures important information about the original set
• It is obtained by applying the valuation map coordinate-wise to the defining equations of the algebraic set and taking the tropical vanishing locus
• The tropicalization process is closely related to the theory of Gröbner bases and initial ideals

Initial ideals

• Given an ideal $I$ in a polynomial ring $k[x_1, \ldots, x_n]$ and a weight vector $w \in \mathbb{R}^n$, the initial ideal $\text{in}_w(I)$ is the ideal generated by the initial forms of the elements of $I$ with respect to $w$
• Initial ideals capture information about the asymptotic behavior of the variety defined by $I$ and are used in the study of Gröbner bases and tropical geometry
• The tropicalization of an algebraic variety can be defined as the set of weight vectors for which the initial ideal contains a monomial

Gröbner bases

• A Gröbner basis is a particular generating set of an ideal with respect to a monomial order, which allows for efficient computation and provides a way to solve systems of polynomial equations
• Gröbner bases are used in the computation of initial ideals and the study of tropical varieties
• The tropicalization of an ideal can be computed using Gröbner bases and the theory of initial ideals

Fundamental theorem of tropical geometry

• The fundamental theorem of tropical geometry states that the tropicalization of an algebraic variety is the support of a polyhedral complex, which is dual to the Gröbner complex of the defining ideal
• This theorem establishes a deep connection between tropical geometry, algebraic geometry, and combinatorics
• It provides a way to study algebraic varieties using combinatorial and polyhedral methods, and conversely, to apply algebraic techniques to combinatorial problems

Tropical semiring

• The tropical semiring is an algebraic structure that underlies tropical geometry, consisting of the real numbers equipped with the operations of maximum and addition
• It is a semiring rather than a ring because the maximum operation does not have an inverse
• The tropical semiring provides a framework for studying tropical polynomials, tropical varieties, and their algebraic properties

Max-plus algebra

• The max-plus algebra is the tropical semiring $(\mathbb{R} \cup \{-\infty\}, \max, +)$, where the addition operation is replaced by maximum and the multiplication operation is replaced by addition
• It is an idempotent semiring, meaning that $\max(x, x) = x$ for all $x$
• The max-plus algebra is used in the study of discrete event systems, optimal control, and tropical geometry

Min-plus algebra

• The min-plus algebra is the tropical semiring $(\mathbb{R} \cup \{\infty\}, \min, +)$, where the addition operation is replaced by minimum and the multiplication operation is replaced by addition
• It is isomorphic to the max-plus algebra via the map $x \mapsto -x$
• The min-plus algebra is used in the study of shortest paths problems, network flows, and tropical geometry

Tropical operations vs classical operations

• Tropical operations (maximum and addition) exhibit different properties compared to their classical counterparts (addition and multiplication)
• For example, the tropical operations are idempotent ($\max(x, x) = x$ and $x + x = x$), while classical addition and multiplication are not
• However, many classical algebraic concepts (polynomials, varieties, intersection theory) have tropical analogues that share similar properties and reveal new connections

Newton polytopes

• Newton polytopes are convex polytopes associated with polynomials, capturing information about the exponent vectors of the monomials appearing in the polynomial
• They play a crucial role in the study of tropical varieties and the relation between algebraic and tropical geometry
• The Newton polytope of a polynomial is closely related to its tropicalization and the combinatorial structure of the associated tropical variety

Definition of Newton polytopes

• Given a polynomial $f(x_1, \ldots, x_n) = \sum_{\alpha} c_{\alpha} x^{\alpha}$, where $\alpha = (\alpha_1, \ldots, \alpha_n) \in \mathbb{Z}_{\geq 0}^n$ and $c_{\alpha} \neq 0$, the Newton polytope of $f$ is the convex hull of the exponent vectors $\alpha$
• The Newton polytope of a polynomial is a lattice polytope, meaning that its vertices have integer coordinates
• Examples include the line segment for linear polynomials, polygons for bivariate polynomials, and higher-dimensional polytopes for multivariate polynomials

Subdivision of Newton polytopes

• A subdivision of a Newton polytope is a collection of smaller polytopes whose union is the original polytope and whose interiors are disjoint
• Subdivisions of Newton polytopes are related to the tropicalization of the polynomial and the combinatorial structure of the associated tropical variety
• The regular subdivisions of the Newton polytope correspond to the different possible initial forms of the polynomial with respect to different weight vectors

Relation to tropical varieties

• The tropicalization of a polynomial is closely related to its Newton polytope
• The tropical hypersurface defined by a polynomial is the corner locus of the piecewise-linear function given by the maximum of the terms in the polynomial, which corresponds to the upper convex hull of the lifted Newton polytope
• The combinatorial structure of the tropical variety associated with a polynomial is dual to the regular subdivision of its Newton polytope induced by the lifting

Tropical compactifications

• Tropical compactifications are a way to extend the theory of tropical varieties to compact spaces, allowing for the study of global properties and intersection theory
• They are obtained by compactifying the tropical affine space using the tropical projective space or other suitable compactifications
• Tropical compactifications provide a framework for studying the relationship between tropical and algebraic geometry in a more global setting

Tropical projective space

• The tropical projective space $\mathbb{TP}^n$ is the quotient of $\mathbb{R}^{n+1}$ by the equivalence relation $(x_0, \ldots, x_n) \sim (x_0 + \lambda, \ldots, x_n + \lambda)$ for any $\lambda \in \mathbb{R}$
• It is a compact space that serves as a natural compactification of the tropical affine space $\mathbb{R}^n$
• Tropical projective space is the target space for the extended tropicalization map and plays a role analogous to classical projective space in algebraic geometry

Extended tropicalization

• The extended tropicalization is a map from a suitable compactification of an algebraic variety (such as the projective space or the toric variety associated with the Newton polytope) to the tropical projective space
• It extends the usual tropicalization map to the compactification, allowing for the study of global properties of the tropicalized variety
• The extended tropicalization is compatible with the tropicalization of subvarieties and the intersection theory on the compactification

Tropical completions

• Tropical completions are another way to compactify tropical varieties, using the theory of valuations and the valuative tree
• They provide a more intrinsic compactification of tropical varieties, without relying on an ambient compactification such as the tropical projective space
• Tropical completions are useful for studying the relationship between tropical varieties and non-archimedean analytic spaces, such as Berkovich spaces

Tropical intersection theory

• Tropical intersection theory is the study of intersections of tropical varieties and the multiplicities arising in these intersections
• It is an important tool for solving enumerative geometry problems and understanding the relationship between tropical and algebraic geometry
• Tropical intersection theory shares many similarities with classical intersection theory, but also exhibits some unique features due to the idempotent nature of the tropical semiring

Stable intersection

• The stable intersection of two tropical varieties is a well-defined tropical cycle that generalizes the notion of set-theoretic intersection
• It is obtained by perturbing the varieties so that their intersection is transverse and then taking the limit of these intersections as the perturbation tends to zero
• The stable intersection is independent of the choice of perturbation and satisfies desirable properties such as associativity and commutativity

Tropical Bézout's theorem

• The tropical Bézout's theorem is an analogue of the classical Bézout's theorem in algebraic geometry, relating the degrees of tropical hypersurfaces to their intersection multiplicity
• It states that the stable intersection of $n$ tropical hypersurfaces in $\mathbb{R}^n$ is a tropical cycle of dimension zero, and its degree is equal to the product of the degrees of the hypersurfaces
• Tropical Bézout's theorem is a powerful tool for solving enumerative geometry problems and understanding the combinatorics of tropical varieties

Tropical Bernstein's theorem

• The tropical Bernstein's theorem is an analogue of the classical Bernstein's theorem in algebraic geometry, relating the mixed volume of Newton polytopes to the number of solutions of a system of polynomial equations
• It states that the stable intersection of $n$ tropical hypersurfaces in $\mathbb{R}^n$ is a tropical cycle of dimension zero, and its degree is equal to the mixed volume of the Newton polytopes of the defining polynomials
• Tropical Bernstein's theorem provides a combinatorial way to compute the number of solutions of a system of polynomial equations and is closely related to the theory of mixed subdivisions of polytopes

Applications of tropicalization

• Tropicalization has found numerous applications in various areas of mathematics, providing new insights and techniques for solving problems
• These applications showcase the power of tropical geometry as a bridge between different fields and highlight the potential for further interdisciplinary research
• Some notable applications of tropicalization include enumerative geometry, phylogenetic trees, and optimization problems

Enumerative geometry

• Enumerative geometry is the study of counting geometric objects satisfying certain conditions, such as the number of curves of a given degree passing through a set of points
• Tropicalization provides a new approach to enumerative geometry problems, by translating them into the language of tropical geometry and using combinatorial techniques to solve them
• Examples include the computation of Gromov-Witten invariants, the enumeration of curves on toric surfaces, and the study of Hurwitz numbers

Phylogenetic trees

• Phylogenetic trees are tree-like structures used in biology to represent the evolutionary relationships among species or other taxa
• Tropicalization has been applied to the study of phylogenetic trees, by considering them as tropical varieties and using tropical geometry techniques to analyze their properties
• This approach has led to new insights into the combinatorics and geometry of phylogenetic trees, as well as the development of efficient algorithms for their construction and comparison

Optimization problems

• Optimization problems involve finding the best solution among a set of feasible options, according to some criteria
• Tropicalization has been used to solve certain classes of optimization problems, by reformulating them in terms of tropical geometry and exploiting the piecewise-linear structure of tropical varieties
• Examples include the tropical version of linear programming, the solution of stochastic games