11.3 Tropical amoebas and Ronkin functions

Tropical amoebas are a key concept in tropical geometry, linking classical algebraic geometry to its tropical counterpart. They're images of algebraic varieties under a logarithmic map, providing a visual and analytical tool for understanding complex geometric structures.

Ronkin functions offer a powerful method for studying amoebas, measuring their size and shape. These functions, along with their associated Monge-Ampère measures, provide crucial insights into the structure and topology of amoebas, bridging complex analysis and tropical geometry.

Definitions of tropical amoebas

• Tropical amoebas are a central object of study in tropical geometry that arise as images of algebraic varieties under a logarithmic map
• Provide a way to visualize and analyze the behavior of algebraic varieties in the tropical setting
• Play a key role in understanding the connections between classical algebraic geometry and tropical geometry

Tropical polynomials and hypersurfaces

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• Tropical polynomials are obtained by replacing the usual arithmetic operations in a polynomial with tropical operations (min for addition, + for multiplication)
• The tropical hypersurface defined by a tropical polynomial consists of the points where the minimum is attained by at least two monomials
• Tropical hypersurfaces are piecewise linear objects that retain important information about the original algebraic variety

Logarithmic limit sets

• The amoeba of an algebraic variety is defined as the image of the variety under a logarithmic map (coordinate-wise logarithm)
• Can be viewed as a limit of rescaled amoebas as a parameter tends to infinity
• Logarithmic limit sets capture the asymptotic behavior of the amoeba and are closely related to the tropical variety

Amoebas vs coamoebas

• Coamoebas are another important object in tropical geometry defined using the argument map instead of the logarithmic map
• While amoebas live in real space, coamoebas live in a compact torus
• The relationship between amoebas and coamoebas provides insights into the structure of the original algebraic variety

Properties of tropical amoebas

• Tropical amoebas exhibit a rich geometric and topological structure that reflects properties of the underlying algebraic variety
• Understanding the properties of amoebas is crucial for studying the behavior of algebraic varieties in the tropical setting
• Many classical results in algebraic geometry have tropical analogues that can be formulated in terms of amoebas

Harnack curves and amoebas

• Harnack curves are real algebraic curves with special topological properties (maximize the number of connected components in the complement of the curve)
• The amoebas of Harnack curves have a particularly nice structure and can be used to construct examples of amoebas with prescribed topological properties
• Harnack curves play a role in the study of real algebraic geometry and have connections to statistical mechanics and other areas

Genus and topology

• The genus of an algebraic curve is a fundamental invariant that measures the complexity of the curve
• The topology of the amoeba of an algebraic curve is closely related to its genus
• Specifically, the number of holes in the amoeba is bounded by the genus of the curve, providing a link between the tropical and classical worlds

Amoebas of lines and hyperplanes

• Lines and hyperplanes are the simplest examples of algebraic varieties, and their amoebas have a particularly simple structure
• The amoeba of a line in the plane is a strip region bounded by two parallel lines
• The amoeba of a hyperplane in higher dimensions is a convex polyhedron
• Amoebas of lines and hyperplanes serve as building blocks for understanding more complex amoebas

Ronkin functions

• Ronkin functions are a powerful tool in the study of amoebas that provide a way to measure the size and shape of an amoeba
• They are named after L. Ronkin who introduced them in the context of complex analysis
• Ronkin functions have found applications in various areas of mathematics, including tropical geometry, complex analysis, and real algebraic geometry

Ronkin function definition

• Given an algebraic variety $V$, the Ronkin function $N_V$ is defined as the Legendre transform of the logarithmic Mahler measure of the defining polynomial of $V$
• Specifically, $N_V(x) = \int_{\log^{-1}(x)} \log|f| d\mu$, where $f$ is the defining polynomial of $V$ and $\mu$ is the Haar measure on the torus
• The Ronkin function can be thought of as a convex function that encodes information about the amoeba of $V$

Monge-Ampère measures

• The Monge-Ampère measure associated with a convex function is a powerful tool in complex analysis and geometry
• In the context of amoebas, the Monge-Ampère measure of the Ronkin function carries important information about the amoeba
• Specifically, the support of the Monge-Ampère measure of $N_V$ coincides with the amoeba of $V$, and the measure describes the distribution of mass within the amoeba

Ronkin functions and amoebas

• The Ronkin function provides a way to recover the amoeba of an algebraic variety from its defining polynomial
• The gradient of the Ronkin function maps the complement of the amoeba to the complement of the coamoeba, establishing a duality between the two objects
• Ronkin functions have been used to prove various results about the structure and topology of amoebas, such as bounds on the number of connected components

Amoeba approximations

• In many applications, it is useful to approximate the amoeba of an algebraic variety with simpler geometric objects
• Amoeba approximations provide a way to study the asymptotic behavior of amoebas and to develop efficient algorithms for amoeba-related problems
• Several types of amoeba approximations have been studied in the literature, each with its own strengths and weaknesses

Piecewise linear approximations

• One natural way to approximate an amoeba is to use a piecewise linear object that captures its main features
• The most common piecewise linear approximation is the spine of the amoeba, which is a tropical variety that can be obtained as a limit of rescaled amoebas
• The spine provides a rough approximation of the shape of the amoeba and can be used to study its combinatorial properties

Convergence of amoeba approximations

• A central question in the study of amoeba approximations is how well they converge to the actual amoeba as the level of approximation increases
• Various notions of convergence have been studied, such as Hausdorff convergence and convergence in the Lebesgue measure
• Understanding the convergence properties of amoeba approximations is important for developing efficient algorithms and for proving theoretical results

Amoeba membership algorithms

• The amoeba membership problem asks whether a given point belongs to the amoeba of an algebraic variety
• This problem arises in various applications, such as solving systems of polynomial equations and studying the topology of real algebraic sets
• Several algorithms have been developed for the amoeba membership problem, some of which rely on amoeba approximations to improve efficiency

Applications of tropical amoebas

• Tropical amoebas have found applications in various areas of mathematics, including complex analysis, real algebraic geometry, and mirror symmetry
• The study of amoebas has led to new results and insights in these fields and has opened up new avenues for research
• Some of the most notable applications of tropical amoebas are highlighted below

Amoebas in complex analysis

• Amoebas first arose in the context of complex analysis as a tool for studying the behavior of analytic varieties near infinity
• The logarithmic limit set of an analytic variety can be viewed as a tropical amoeba, providing a connection between complex analysis and tropical geometry
• Amoebas have been used to prove results in complex analysis, such as the existence of certain types of holomorphic functions

Amoebas in real algebraic geometry

• Real algebraic geometry studies the properties of real algebraic varieties, which are defined by polynomial equations with real coefficients
• Amoebas provide a way to study the topology and geometry of real algebraic varieties using techniques from tropical geometry
• In particular, the complement of an amoeba can be used to study the number of connected components of a real algebraic variety

Amoebas in mirror symmetry

• Mirror symmetry is a profound duality between complex geometry and symplectic geometry that has far-reaching implications in mathematics and physics
• Tropical geometry has emerged as a key tool in the study of mirror symmetry, and amoebas play a central role in this connection
• In particular, the amoeba of a variety in one side of the mirror can be used to construct the mirror variety on the other side, providing a concrete realization of the mirror map