🌴Tropical Geometry Unit 2 – Tropical Polynomials and Curves

Key Concepts and Definitions

• Tropical geometry studies geometric objects defined by polynomial equations using tropical algebra operations
• Tropical algebra replaces classical arithmetic operations with tropical addition (maximum) and tropical multiplication (ordinary addition)
• Tropical semiring $(\mathbb{R} \cup \{-\infty\}, \oplus, \odot)$ where $a \oplus b = \max(a,b)$ and $a \odot b = a + b$
• Tropical polynomials are polynomials with coefficients in the tropical semiring and variables raised to non-negative integer powers
  • Example: $f(x) = 3 \odot x^2 \oplus 1 \odot x \oplus 5 = \max(3 + 2x, 1 + x, 5)$
• Tropical curves are the geometric objects defined by tropical polynomial equations in the plane or higher-dimensional spaces
• Newton polygon of a tropical polynomial encodes its monomials and provides insights into the structure of the corresponding tropical curve
• Puiseux series are power series with rational exponents used to study the local behavior of tropical curves near a point

Tropical Algebra Basics

• Idempotency of tropical addition: $a \oplus a = a$ for all $a \in \mathbb{R} \cup \{-\infty\}$
• Tropical multiplication distributes over tropical addition: $a \odot (b \oplus c) = (a \odot b) \oplus (a \odot c)$
• No additive inverses in the tropical semiring, only a multiplicative identity (0) and absorbing element ($-\infty$)
• Tropical division corresponds to subtraction: $a \oslash b = a - b$ for $a, b \in \mathbb{R}$
• Tropical powers: $a^{\odot n} = na$ for $a \in \mathbb{R}$ and $n \in \mathbb{N}$
• Tropical polynomial evaluation: substitute values for variables and compute using tropical operations
  • Example: $f(x) = 2 \odot x^2 \oplus 3 \odot x \oplus 1$, then $f(4) = \max(2 + 2 \cdot 4, 3 + 4, 1) = 11$

Tropical Polynomials: Structure and Properties

• Tropical polynomials are piecewise-linear functions, with each piece corresponding to a monomial
• The graph of a tropical polynomial consists of line segments and rays, forming a tropical curve
• Roots of a tropical polynomial are the points where the maximum is attained by at least two monomials
  • Example: $f(x) = 2 \odot x^2 \oplus 3 \odot x \oplus 1$, roots are $x = -1$ and $x = 1$
• Factorization of tropical polynomials: $f(x) = \bigoplus_{i=1}^n (a_i \odot x \oplus b_i)$ where $a_i, b_i \in \mathbb{R}$
• Fundamental Theorem of Tropical Algebra: every tropical polynomial can be uniquely factored into linear factors
• Newton polygon and subdivision determine the shape and combinatorial structure of the tropical curve
• Valuations and Puiseux series expansions describe the local behavior of tropical curves near a point

Tropical Curves: Visualization and Analysis

• Tropical curves are piecewise-linear graphs in the plane or higher-dimensional spaces
• Dual subdivision of the Newton polygon provides a combinatorial description of the tropical curve
  • Vertices correspond to connected components, edges to line segments, and rays to unbounded edges
• Genus of a tropical curve is determined by the number of interior lattice points in the Newton polygon
• Intersection theory for tropical curves: stable intersection, multiplicities, and Bézout's theorem
• Tropical Bézout's theorem: the number of intersections (counting multiplicities) of two tropical curves is the product of their degrees
• Riemann-Roch theorem for tropical curves relates the genus, degree, and dimension of the space of rational functions
• Tropical modifications alter the Newton polygon and change the structure of the tropical curve

Applications in Optimization and Modeling

• Tropical geometry provides a framework for solving optimization problems, particularly those involving piecewise-linear functions
• Shortest paths and network flow problems can be formulated using tropical algebra
  • Example: finding the shortest path in a weighted graph using the tropical semiring
• Scheduling problems, such as project management and resource allocation, have tropical geometric interpretations
• Phylogenetic trees and models of evolution can be studied using tropical geometry
  • Inferring evolutionary relationships and estimating mutation rates using tropical methods
• Auction theory and mechanism design benefit from tropical geometric techniques
• Tropical linear programming extends classical linear programming to the tropical semiring
• Game theory and decision-making under uncertainty can be analyzed using tropical methods

Computational Techniques and Tools

• Gröbner bases for tropical ideals provide a computational framework for studying tropical varieties
• Tropical Gröbner bases can be computed using the Buchberger algorithm adapted to the tropical semiring
• Tropical Singular is a software package for computations in tropical geometry, built on the Singular computer algebra system
• Polymake is a software for polytopes and polyhedra, with functionality for tropical geometry
• Gfan is a software package for computing Gröbner fans and tropical varieties
• Tropical.jl is a Julia package for tropical arithmetic, polynomials, and curves
• TropicalGeometry package in Macaulay2 provides tools for studying tropical curves and hypersurfaces
• Visualization of tropical curves can be done using packages like Matplotlib (Python) or ggplot2 (R)

Advanced Topics and Current Research

• Tropical Grassmannians and tropical Plücker vectors for studying higher-dimensional tropical varieties
• Tropical Hilbert functions and Hilbert polynomials for understanding the structure of tropical ideals
• Tropical toric varieties and their connections to classical toric varieties and convex polytopes
• Tropical moduli spaces, such as the moduli space of tropical curves, and their applications in enumerative geometry
• Tropical mirror symmetry relates tropical geometry to complex geometry and string theory
• Tropical Hodge theory and tropical cohomology for studying the topology of tropical varieties
• Tropical dynamical systems and their applications in modeling biological and physical systems
• Tropical representation theory and its connections to classical representation theory and combinatorics

Practice Problems and Examples

1. Compute the tropical product $(3 \odot x^2 \oplus 1 \odot x \oplus 4) \odot (2 \odot x \oplus 5)$.
2. Find the roots of the tropical polynomial $f(x) = 2 \odot x^3 \oplus 3 \odot x^2 \oplus 1 \odot x \oplus 4$.
3. Determine the Newton polygon of the tropical polynomial $f(x, y) = 3 \odot x^2 \odot y \oplus 2 \odot x \odot y^2 \oplus 1 \odot x \oplus 4 \odot y \oplus 5$.
4. Sketch the tropical curve defined by the polynomial $f(x, y) = 2 \odot x^2 \oplus 3 \odot y^2 \oplus 1 \odot x \odot y \oplus 4 \odot x \oplus 5 \odot y \oplus 6$.
5. Find the shortest path between two vertices in a weighted graph using tropical algebra.
6. Compute the Gröbner basis of a tropical ideal using the Buchberger algorithm.
7. Determine the genus of a tropical curve given its Newton polygon.
8. Analyze a tropical dynamical system and identify its fixed points and periodic orbits.