Tropical hypersurfaces are fascinating geometric objects that arise from tropical polynomials. They provide a bridge between algebra and geometry, offering insights into the structure of classical algebraic varieties through a combinatorial lens.
These piecewise linear objects are defined as the set of points where a attains its maximum. By studying their properties, we can uncover deep connections between tropical geometry and classical algebraic geometry, shedding light on complex mathematical relationships.
Definition of tropical hypersurfaces
A is the set of points in tropical projective space where a tropical polynomial attains its maximum
Tropical hypersurfaces are piecewise linear objects that arise as limits of classical algebraic varieties over fields with valuation
Studying tropical hypersurfaces provides insights into the combinatorial structure of algebraic varieties and their degenerations
Tropical polynomials
Tropical polynomials are polynomials where the usual arithmetic operations are replaced by tropical operations: addition is replaced by maximum and multiplication is replaced by usual addition
The coefficients of tropical polynomials are elements of the tropical semiring, which is the real numbers together with negative infinity
Tropical polynomials can be used to define tropical hypersurfaces and study their geometric and combinatorial properties
Support of tropical polynomials
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The support of a tropical polynomial is the set of exponent vectors of the monomials appearing in the polynomial with a coefficient not equal to negative infinity
The convex hull of the support is the of the tropical polynomial
The support of a tropical polynomial determines the combinatorial type of the corresponding tropical hypersurface
Coefficients in tropical polynomials
The coefficients in a tropical polynomial determine the position of the tropical hypersurface in the tropical projective space
Changing the coefficients of a tropical polynomial can lead to different combinatorial types of the corresponding tropical hypersurface
The coefficients of a tropical polynomial can be used to study the moduli space of tropical hypersurfaces of a given degree
Tropical hypersurface as corner locus
A tropical hypersurface can be viewed as the corner locus of a convex piecewise-linear function, i.e., the set of points where the function is not differentiable
This interpretation allows for a geometric understanding of tropical hypersurfaces and their properties
The corner locus perspective is useful for studying the relationship between tropical hypersurfaces and their dual subdivisions
Convex piecewise-linear functions
A convex piecewise-linear function is a function that is the maximum of a finite number of affine linear functions
The domains of linearity of a convex piecewise-linear function form a polyhedral subdivision of the space
Convex piecewise-linear functions are closely related to tropical polynomials and can be used to define tropical hypersurfaces
Tropical hypersurface as non-differentiable locus
The tropical hypersurface associated with a tropical polynomial f is the set of points where the convex piecewise-linear function max{ai+⟨x,i⟩} is not differentiable, where ai are the coefficients of f and i runs over the support of f
The non-differentiability of the convex piecewise-linear function occurs along the edges and vertices of the polyhedral subdivision induced by the function
This characterization of tropical hypersurfaces allows for a geometric and combinatorial study of their properties
Newton subdivision of tropical hypersurfaces
The of a tropical hypersurface is a polyhedral subdivision of the Newton polytope of the associated tropical polynomial
Each cell in the Newton subdivision corresponds to a domain of linearity of the convex piecewise-linear function defining the tropical hypersurface
The Newton subdivision encodes important combinatorial information about the tropical hypersurface, such as its intersection with other hypersurfaces
Definition of Newton subdivision
The Newton subdivision of a tropical polynomial f is the polyhedral subdivision of the Newton polytope of f induced by the convex piecewise-linear function max{ai+⟨x,i⟩}, where ai are the coefficients of f and i runs over the support of f
The cells of the Newton subdivision are the projections of the domains of linearity of the convex piecewise-linear function onto the Newton polytope
The Newton subdivision can be computed using the regular subdivision induced by the coefficients of the tropical polynomial
Cells in Newton subdivision
The cells in the Newton subdivision of a tropical hypersurface correspond to the components of the complement of the hypersurface in the tropical projective space
The dimension of a cell in the Newton subdivision is equal to the codimension of the corresponding component of the complement of the tropical hypersurface
The cells in the Newton subdivision can be used to study the topology and combinatorics of the tropical hypersurface
Dual graph of Newton subdivision
The dual graph of the Newton subdivision of a tropical hypersurface is a graph where each vertex corresponds to a cell in the subdivision and two vertices are connected by an edge if the corresponding cells share a facet
The dual graph encodes the adjacency relations between the components of the complement of the tropical hypersurface
The dual graph can be used to study the connectivity and other properties of the tropical hypersurface
Balancing condition
The is a local condition on the coefficients of a tropical polynomial that ensures the smoothness of the corresponding tropical hypersurface
It states that for each facet of the Newton subdivision, the sum of the lattice lengths of the edges adjacent to the facet, weighted by their corresponding coefficients, vanishes
The balancing condition is a tropical analogue of the smoothness condition for algebraic varieties
Statement of balancing condition
Let f be a tropical polynomial and let τ be a facet of the Newton subdivision of f. For each edge e adjacent to τ, let ve be the primitive integer vector along e pointing away from τ and let ae be the coefficient of the monomial corresponding to the endpoint of e opposite to τ. The balancing condition states that ∑eaeve=0
The balancing condition can be checked locally at each facet of the Newton subdivision
If the balancing condition is satisfied, the tropical hypersurface is called smooth or balanced
Balancing condition vs smoothness
The balancing condition is a necessary but not sufficient condition for the smoothness of a tropical hypersurface
A tropical hypersurface is smooth if and only if it is locally the graph of a convex piecewise-linear function and the balancing condition is satisfied at each facet of the Newton subdivision
Smoothness of tropical hypersurfaces is important for studying their intersection theory and relationship with classical algebraic geometry
Tropical Bézout's theorem
Tropical Bézout's theorem is a fundamental result in tropical geometry that relates the intersection of tropical hypersurfaces to the mixed volume of their Newton polytopes
It states that the number of points of n tropical hypersurfaces in n-dimensional tropical projective space, counted with multiplicities, is equal to the mixed volume of their Newton polytopes
Tropical Bézout's theorem is a powerful tool for studying the intersection theory of tropical varieties and their relationship with classical algebraic geometry
Classical Bézout's theorem
Classical Bézout's theorem states that the number of intersection points of n algebraic hypersurfaces in n-dimensional projective space, counted with multiplicities, is equal to the product of their degrees
It is a fundamental result in classical algebraic geometry and has numerous applications in and other areas
Tropical Bézout's theorem can be seen as a generalization of classical Bézout's theorem to the tropical setting
Statement of tropical Bézout's theorem
Let f1,…,fn be tropical polynomials defining tropical hypersurfaces in n-dimensional tropical projective space. The number of stable intersection points of these hypersurfaces, counted with multiplicities, is equal to the mixed volume MV(P1,…,Pn) of their Newton polytopes P1,…,Pn
The mixed volume is a combinatorial invariant of the Newton polytopes that generalizes the notion of volume to the multivariate setting
Tropical Bézout's theorem holds for any choice of coefficients of the tropical polynomials, as long as the intersection is stable
Intersection multiplicity in tropical Bézout's theorem
The intersection multiplicity of a stable intersection point of tropical hypersurfaces is a local invariant that reflects the combinatorial structure of the intersection
It can be computed using the balancing condition and the lattice lengths of the edges adjacent to the intersection point in the Newton subdivisions of the hypersurfaces
The sum of the intersection multiplicities over all stable intersection points is equal to the mixed volume of the Newton polytopes, as stated in tropical Bézout's theorem
Stable intersection of tropical hypersurfaces
A stable intersection of tropical hypersurfaces is an intersection that is preserved under small perturbations of the coefficients of the defining tropical polynomials
Stable intersections are the tropical analogue of transversal intersections in classical algebraic geometry
Studying stable intersections is crucial for developing a well-defined intersection theory for tropical varieties
Definition of stable intersection
An intersection point of n tropical hypersurfaces in n-dimensional tropical projective space is called stable if it is isolated and the intersection multiplicity remains constant under small perturbations of the coefficients of the defining tropical polynomials
Equivalently, an intersection point is stable if the corresponding cells in the Newton subdivisions of the hypersurfaces intersect transversely and the balancing condition is satisfied at the intersection point
Stable intersections are the key ingredient in the formulation of tropical Bézout's theorem and other results in theory
Stable intersection vs transversal intersection
In classical algebraic geometry, a transversal intersection of hypersurfaces is an intersection where the tangent spaces of the hypersurfaces at the intersection point span the ambient space
Stable intersections in tropical geometry can be seen as a combinatorial analogue of transversal intersections, where the role of tangent spaces is played by the cells in the Newton subdivisions
While transversal intersections are generic in classical algebraic geometry, stable intersections are not necessarily generic in tropical geometry and may require specific choices of coefficients
Examples of tropical hypersurfaces
Tropical hypersurfaces arise in various contexts and have interesting geometric and combinatorial properties
Studying examples of tropical hypersurfaces helps develop intuition and showcases the rich structure of tropical varieties
Some notable examples include linear tropical hypersurfaces, quadratic tropical hypersurfaces, and tropical hypersurfaces in higher dimensions
Linear tropical hypersurfaces
A linear tropical hypersurface is the tropical vanishing locus of a tropical linear polynomial, i.e., a polynomial of the form max{a1+x1,…,an+xn}
Linear tropical hypersurfaces are tropical hyperplanes and divide the tropical projective space into n+1 regions corresponding to the domains of linearity of the tropical polynomial
The combinatorial structure of a linear tropical hypersurface is determined by the arrangement of the coefficients a1,…,an
Quadratic tropical hypersurfaces
A quadratic tropical hypersurface is the tropical vanishing locus of a tropical quadratic polynomial, i.e., a polynomial of the form max{aij+xi+xj}, where the maximum is taken over a subset of pairs (i,j)
Quadratic tropical hypersurfaces are also known as tropical conics and have a rich combinatorial structure determined by the coefficients aij
The Newton subdivision of a quadratic tropical hypersurface is a subdivision of the triangle with vertices (0,0), (2,0), and (0,2), and the dual graph is a planar graph
Tropical hypersurfaces in higher dimensions
Tropical hypersurfaces in higher dimensions are defined by tropical polynomials in more than two variables and exhibit a wide range of combinatorial and geometric phenomena
The Newton subdivisions of higher-dimensional tropical hypersurfaces are subdivisions of higher-dimensional polytopes and can have intricate combinatorial structures
Higher-dimensional tropical hypersurfaces arise in various applications, such as the study of amoebas of algebraic varieties, the of moduli spaces, and the geometry of toric varieties