We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem \cite{K-00} in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces defined by a maximally sparse polynomial and coamoebas of maximally sparse complex tropical hypersurfaces.
Let $V$ be a complex algebraic hypersurface defined by a polynomial $f$ with Newton polytope $Δ$. It is well known that the spine of its amoeba has a structure of a tropical hypersurface. We prove in this paper that there exists a complex tropical hypersurface $V_{\infty, f}$ such that its coamoeba is homeomorphic to the closure in the real torus of the coamoeba of $V$. Moreover, the coamoeba of $V_{\infty, f}$ contains an arrangement of $(n-1)$-torus depending only on the geometry of $Δ$ and the coefficients of $f$. In addition, we can consider this arrangement, as a weighted codual hyperplan…
A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. Similarly, a non-archimedean coamoeba is the image of a subvariety of a torus over a non-archimedean field with complex residue field under an argument map. The phase tropical variety is the closure of the image under the pair of maps, tropicalization and argument. We describe the structure of non-archimedean coamoebae and phaseropical varieties in terms of complex coamoebae and their phase limit sets. The argument map depends upon a section of the valuation map, and we explain how this choice …
We introduce the notion of a tropical coamoeba which gives a combinatorial description of the Fukaya category of the mirror of a toric Fano stack. We show that the polyhedral decomposition of a real n-torus into (n + 1) permutohedra gives a tropical coamoeba for the mirror of the projective space, and prove a torus-equivariant version of homological mirror symmetry for the projective space. As a corollary, we obtain homological mirror symmetry for toric orbifolds of the projective space.
Given a hypersurface coamoeba of a Laurent polynomial f, it is an open problem to describe the structure of its set of connected complement components. In this paper we approach this problem by introducing the lopsided coamoeba. We show that the closed lopsided coamoeba comes naturally equipped with an order map, i.e. a map v from its set of connected complement components to a translated lattice inside the zonotope of a Gale dual of the point configuration supp(f). Under a natural assumption, the map v is a bijection. Finally we use this map to obtain new results concerning coamoebas of polyn…
First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that phase tropical hypersurfaces with smooth tropicalization, possess naturally a smooth differentiable structure. Moreover, we prove that phase tropical hypersurfaces possess a natural symplectic structure.
This paper is devoted to the bounding and computation of the dimension of deformation spaces of tropical curves and hypersurfaces. This characteristic is interesting in light of the fact that it often coincides with the dimension of equisingular (equigeneric etc.) deformation spaces of algebraic curves and hypersurfaces. In this paper, we obtain a series of precise formulas, upper and lower bounds, and algorithms for computing dimension of deformation spaces of various classes of tropical curves and hypersurfaces.
We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless their stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize to them and the tropicalization of their intersection.
Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.
The concepts of tropical-semiring and tropical hypersurface, are extended for an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator "tropicalization" we conclude with an extension of Kapranov's theorem to algebraically closed fields together with a valuation over an ordered group.
We study the notion of singular tropical hypersurfaces of any dimension. We characterize the singular points in terms of tropical Euler derivatives and we give an algorithm to compute all singular points. We also describe non-transversal intersection points of planar tropical curves.
This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real tropical hypersurfaces. If V is a tropical hypersurface defined over the field of real Puiseux series, it has a real part RV which is a polyhedral complex. We define the total curvature of V (resp. RV) by using the total curvature of Amoebas and passing to the limit. We also define the "polyhedral total curvature" of the real part RV of a generic tropical hypersurface. The main results we prove about thes…
Given a projective variety $X$ of codimension $k+1$ in $\mathbb{P}^n$ the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Tro…
We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We give a geometric/combinatorial way and a tropical linear-algebraic way to approach this question. First, given a fixed hypersurface, we show how one can determine whether points on the hypersurface determine it by looking at a corresponding marking of the dual complex. With a regularity condition on the dual complex and when the number of points is minimal, we …
We produce for each tropical hypersurface $V(ϕ)\subset Q=\mathbb{R}^n$ a Lagrangian $L(ϕ)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(ϕ)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror.
We prove a bound conjectured by Itenberg on the Betti numbers of real algebraic hypersurfaces near non-singular tropical limits. These bounds are given in terms of the Hodge numbers of the complexification. To prove the conjecture we introduce a real variant of tropical homology and define a filtration on the corresponding chain complex inspired by Kalinin's filtration. The spectral sequence associated to this filtration converges to the homology groups of the real algebraic variety and we show that the terms of the first page are tropical homology groups with $\mathbb{Z}_2$-coefficients. The …
We prove that a smooth tropical hypersurface in $\mathbb{R}^3$ can be lifted to a smooth embedded Lagrangian submanifold in $(\mathbb{C}^*)^3$. This completes the proof of the result announced in the article "Lagrangian pairs pants" arXiv:1802.02993. The idea of the proof is to use Lagrangian pairs of pants as the main building blocks.
We develop a version of Mikhalkin's lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $δ$ dimensional linear space of degree $ d $ real hypersurfaces containing $\frac{1}{δ!} (γ_nd^n)^δ+ O(d^{nδ-1})$ hypersurfaces with $ δ$ real nodes, where $ γ_n $ are positive and given by a recursive formula. This is asymptotically comparable to the number $ \frac{1}{δ!} \left( (…
We call a real algebraic hypersurface in $(\mathbb{C}^*)^n$ simplicial if it is given by a real Laurent polynomial in $n$-variables that has exactly $n+1$ monomials with non-zero coefficients and such that the convex hull in $\mathbb{R}^n$ of the $n+1$ points of $\mathbb{Z} ^n$ corresponding to the exponents is a non-degenerate $n$-dimensional simplex. Such hypersurfaces are natural building blocks from which more complicated objects can be constructed, for example using O. Viro's Patchworking method. Drawing inspiration from related work by G. Kerr and I. Zharkov, we describe the action of th…
We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $\mathbb{P}^2$ or a surface in $\mathbb{P}^3$, we provide an explicit description of $\text{Trop}(X^*)$ in terms of $\text{Trop}(X)$, as long as $\text{Trop}(X)$ is smooth and satisfies a mild genericity condition. As a consequence, when $X$ is a curve we describe the transformation of Newton polygons under projective duality, and recover classical formulas for the degree of a dual plane curve. For higher dimensional hypersurfaces $X$, we give a partial description of $\text{Trop}(X^*)$.
We explore the concept of real tropical basis of an ideal in the field of real Puiseux series. We show explicit tropical bases of zero-dimensional real radical ideals, linear ideals and hypersurfaces coming from combinatorial patchworking. But we also show that there exist real radical ideals that do not admit a tropical basis. As an application, we show how to compute the set of singular points of a real tropical hypersurface. i.e. we compute the real tropical discriminant.
A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected.
We define nondegenerate tropical complete intersections imitating the corresponding definition in complex algebraic geometry. As in the complex situation, all nonzero intersection multiplicity numbers between tropical hypersurfaces defining a nondegenerate tropical complete intersection are equal to 1. The intersection multiplicity numbers we use are sums of mixed volumes of polytopes which are dual to cells of the tropical hypersurfaces. We show that the Euler characteristic of a real nondegenerate tropical complete intersection depends only on the Newton polytopes of the tropical polynomials…
We consider the residual B-model variation of Hodge structure of Iritani defined by a family of toric Calabi--Yau hypersurfaces over a punctured disk $D \setminus \{ 0\}$. It is naturally extended to a logarithmic variation of polarized Hodge structure of Kato--Usui on the whole disk $D$. By restricting it to the origin, we obtain a polarized logarithmic Hodge structure (PLH) on the standard log point. In this paper, we describe the PLH in terms of the integral affine structure of the dual intersection complex of the toric degeneration in the Gross--Siebert program.
We present our implementation of an algorithm which functions as a numerical oracle for the Newton polytope of a hypersurface in the Macaulay2 package NumericalNP.m2. We propose a tropical membership test, relying on this algorithm, for higher codimension varieties based on ideas from Hept and Theobald. To showcase this software, we investigate the Newton polytope of both a hypersurface coming from algebraic vision and the Luroth invariant.
An ideal $I$ is said to be "well-poised" if all of the initial ideals obtained from points in the tropical variety $Trop(I)$ are prime. This condition was first defined by Nathan Ilten and the third author. We classify all well-poised hypersurfaces over an algebraically closed field. We also study the tropical varieties and associated Newton-Okounkov bodies of these hypersurfaces.
We study singularities in tropical hypersurfaces defined by a valuation over a field of positive characteristic. We provide a method to compute the set of singular points of a tropical hypersurface in positive characteristic and the p-adic case. This computation is applied to determine all maximal cones of the tropical linear space of univariate polynomials of degree $n$ and characteristic $p$ with a fixed double root and the fan of all tropical polynomials that have $0$ as a double root independently of the characteristic. We also compute, by pure tropical means, the number of vertices, edges…
The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be…
We report on a recent implementation of patchworking and real tropical hypersurfaces in $\texttt{polymake}$. As a new mathematical contribution we provide a census of Betti numbers of real tropical surfaces.
Let $\{V_q\}_{q}$ be a complex one-parameter family of smooth hypersurfaces in a toric variety. In this paper, we give a concrete description of the monodromy transformation of $\{V_q\}_q$ around $q=\infty$ in terms of tropical geometry. The main tool is the tropical localization introduced by Mikhalkin.
Given any complex Laurent polynomial $f$, $\mathrm{Amoeba}(f)$ is the image of its complex zero set under the coordinate-wise log absolute value map. We give an efficiently constructible polyhedral approximation, $\mathrm{ArchtTrop}(f)$, of $\mathrm{Amoeba}(f)$, and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of $f$, for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in $\mathrm{ArchTrop}(f)$ is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for $\mathrm{…
Applying tropical geometry a framework for mirror symmetry, including a mirror construction for Calabi-Yau varieties, was proposed by the author. We discuss the conceptual foundations of this construction based on a natural mirror map identifying deformations and divisors. We show how the construction specializes to that by Batyrev for hypersurfaces and its generalization by Batyrev and Borisov to complete intersections. Based on an explicit example we comment on the implementation in the Macaulay2 package SRdeformations.
Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as t…
Kapranov's theorem is a foundational result in tropical geometry. It states that the set of tropicalisations of points on a hypersurface coincides precisely with the tropical variety of the tropicalisation of the defining polynomial. The aim of this paper is to generalise Kapranov's theorem, replacing the role of a valuation map, from a field to the real numbers union negative infinity, with a more general class of hyperfield homomorphisms, whose target is the tropical hyperfield and satisfy a relative algebraic closure condition. To provide an example of such a hyperfield homomorphism, the ma…
