2.1 Tropical polynomial functions

Tropical polynomial functions are a key concept in tropical geometry, using unique arithmetic operations like min for addition and regular addition for multiplication. They generalize classical polynomials, offering a new way to model and solve problems in various fields.

These functions can be univariate or multivariate, and their evaluation differs from classical polynomials. Tropical polynomials have interesting properties, including roots, Newton polygons, and factorization, which provide powerful tools for analysis and problem-solving in optimization and modeling.

Tropical polynomial definition

• Tropical polynomials are a fundamental concept in tropical geometry that generalizes the notion of classical polynomials
• They are defined using the tropical semiring, which has different arithmetic operations compared to the classical ring of polynomials

Tropical addition and multiplication

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• Tropical addition is defined as the minimum operation, denoted as $\oplus$, where $a \oplus b = \min(a, b)$
• Tropical multiplication is defined as the usual addition, denoted as $\otimes$, where $a \otimes b = a + b$
• These operations form the basis for constructing tropical polynomials and performing computations on them

Formal definition

• A tropical polynomial $f(x)$ in one variable is defined as $f(x) = \bigoplus_{i=0}^{n} a_i \otimes x^{\otimes i} = \min_{0 \leq i \leq n}(a_i + ix)$, where $a_i$ are the coefficients and $x$ is the variable
• The degree of a tropical polynomial is the highest exponent of the variable appearing in the polynomial

Univariate vs multivariate

• Univariate tropical polynomials involve only one variable, such as $f(x) = \min(3 + 2x, 1 + 3x, 4x)$
• Multivariate tropical polynomials involve multiple variables, such as $f(x, y) = \min(2 + x + y, 3 + 2x, 1 + 2y)$
• Multivariate tropical polynomials are used to model more complex systems and relationships in tropical geometry

Tropical polynomial evaluation

• Evaluating tropical polynomials involves computing the value of the polynomial at a given point using tropical arithmetic operations
• The process differs slightly for univariate and multivariate polynomials

Evaluating univariate polynomials

• To evaluate a univariate tropical polynomial $f(x)$ at a point $x = a$, substitute $a$ for $x$ and perform the tropical operations
• For example, if $f(x) = \min(3 + 2x, 1 + 3x, 4x)$ and $a = 2$, then $f(2) = \min(3 + 2 \cdot 2, 1 + 3 \cdot 2, 4 \cdot 2) = \min(7, 7, 8) = 7$

Evaluating multivariate polynomials

• Evaluating a multivariate tropical polynomial follows a similar process, substituting values for each variable and performing tropical operations
• For example, if $f(x, y) = \min(2 + x + y, 3 + 2x, 1 + 2y)$ and $(a, b) = (1, 2)$, then $f(1, 2) = \min(2 + 1 + 2, 3 + 2 \cdot 1, 1 + 2 \cdot 2) = \min(5, 5, 5) = 5$

Tropical polynomial as piecewise linear function

• A tropical polynomial can be viewed as a piecewise linear function when plotted in the Euclidean plane
• The graph of a tropical polynomial consists of line segments, each corresponding to a monomial term in the polynomial
• The minimum operation in the tropical semiring results in the selection of the line segment with the smallest value at each point

Roots of tropical polynomials

• Roots of tropical polynomials are points where the polynomial achieves its minimum value
• They have a graphical interpretation and can have different multiplicities

Definition of roots

• A root of a tropical polynomial $f(x)$ is a value $r$ such that $f(r) = \min_{0 \leq i \leq n}(a_i + ir)$ is attained at least twice
• In other words, at least two monomials in the polynomial achieve the minimum value at the root

Graphical interpretation of roots

• Graphically, roots of a tropical polynomial correspond to the intersection points of the line segments in its piecewise linear representation
• The $x$-coordinates of these intersection points are the roots of the polynomial

Multiplicity of roots

• The multiplicity of a root is the number of monomials that achieve the minimum value at that point minus one
• For example, if three monomials attain the minimum value at a root, its multiplicity is 2
• Higher multiplicity roots indicate a more significant intersection point in the tropical polynomial's graph

Newton polygon of tropical polynomial

• The Newton polygon is a geometric object associated with a tropical polynomial that encodes information about its roots and factorization
• It is constructed using the exponents and coefficients of the polynomial

Definition of Newton polygon

• The Newton polygon of a tropical polynomial $f(x) = \bigoplus_{i=0}^{n} a_i \otimes x^{\otimes i}$ is the convex hull of the points $(i, a_i)$ in the Euclidean plane
• Each point in the Newton polygon corresponds to a monomial term in the polynomial

Constructing Newton polygon

• To construct the Newton polygon, plot the points $(i, a_i)$ for each monomial term in the polynomial
• Take the convex hull of these points, which is the smallest convex polygon containing all the points
• The edges of the convex hull form the Newton polygon

Properties of Newton polygon

• The slopes of the edges of the Newton polygon are related to the roots of the tropical polynomial
• The negative reciprocals of the slopes give the roots of the polynomial
• The length of an edge corresponds to the multiplicity of the associated root

Newton polygon vs polynomial roots

• The Newton polygon provides a visual representation of the roots and their multiplicities
• It allows for a quick determination of the roots without explicitly solving the polynomial
• Changes in the Newton polygon reflect changes in the roots and factorization of the polynomial

Factorization of tropical polynomials

• Factorization of tropical polynomials is the process of expressing a polynomial as a product of irreducible factors
• Tropical polynomials have unique factorization properties

Irreducible tropical polynomials

• An irreducible tropical polynomial is one that cannot be expressed as the product of two non-constant polynomials
• Irreducible polynomials are the building blocks for factorization

Unique factorization theorem

• The unique factorization theorem states that every tropical polynomial can be uniquely factored into a product of irreducible polynomials, up to scaling and reordering
• This property is analogous to the fundamental theorem of algebra for classical polynomials

Factorization algorithms

• There are several algorithms for factoring tropical polynomials, such as the Newton polygon method and the tropical Hensel lifting
• The Newton polygon method uses the edges of the Newton polygon to determine the irreducible factors
• Tropical Hensel lifting is an iterative algorithm that lifts factorizations from the residue field to the polynomial ring

Applications of tropical polynomials

• Tropical polynomials have various applications in mathematics and other fields
• They are used in solving optimization problems, interpolation, and modeling systems of equations

Tropical polynomial interpolation

• Tropical polynomial interpolation involves finding a tropical polynomial that passes through a given set of points
• It has applications in data fitting and approximation problems
• Techniques such as tropical Lagrange interpolation and Newton interpolation are used

Tropical polynomial optimization

• Tropical polynomials can be used to solve optimization problems, particularly in discrete event systems and scheduling
• The piecewise linear nature of tropical polynomials allows for efficient optimization algorithms
• Examples include the max-plus algebra used in modeling transportation networks and production systems

Tropical polynomial systems

• Systems of tropical polynomial equations arise in various contexts, such as in the study of tropical varieties
• Solving these systems involves finding common roots or intersections of multiple tropical polynomials
• Techniques from tropical algebraic geometry, such as tropical elimination theory and tropical basis computation, are employed in solving these systems