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We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization A ⊥ and the double tropical orthogonalization A ⊥ ⊥ of a subset A of the vector space (R ∪ { ∞ }) n . We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety. [ABSTRACT FROM AUTHOR]

We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical varieties, using intersections with affine hyperplanes to reduce the dimension to zero. We use the algorithms to show that the tropical Grassmannians G3,8 and G4,8 are not simplicial. [ABSTRACT FROM AUTHOR]

Given any complex Laurent polynomial f , Amoeba ( f ) is the image of its complex zero set under the coordinate-wise log absolute value map. We discuss an efficiently constructible polyhedral approximation, ArchTrop ( f ) , of 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 ArchTrop ( f ) is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for Amoeba ( f ) , which is NP -hard already in one variable. ArchTrop ( f ) can thus serve as a canonical low order approximation to start a higher order iterative polynomial system solving algorithm, e.g., homotopy continuation. [ABSTRACT FROM AUTHOR]

We generalise the notion of a Gröbner fan to ideals in R 〚 t 〛 [ x 1 , … , x n ] for certain classes of coefficient rings R and give a constructive proof that the Gröbner fan is a rational polyhedral fan. For this we introduce the notion of initially reduced standard bases and show how these can be computed in finite time. We deduce algorithms for computing the Gröbner fan, implemented in the computer algebra system Singular . The problem is motivated by the wish to compute tropical varieties over the p -adic numbers. [ABSTRACT FROM AUTHOR]

Letkbe a field with a real valuationνandRak-algebra. We show that there exist ak-algebraKand a valuationμonKextendingνsuch that any real valuation ofRis induced byμthrough some homomorphism fromRtoK. Letνbe trivial andRa complete local Noetherian ring with the residue fieldk. LetKbe the ringof Hahn series with its natural valuationμand coefficients in. We prove the following weak universality property: For any local valuationvand a finite set of elementsofR, there exists a homomorphismf:R→Ksuch that,i = 1,…,n. This implies that iffor an idealI, then every point of the local tropicalization ofIlifts to aK-point ofR. [ABSTRACT FROM PUBLISHER]

We study tropical commuting matrices from two viewpoints: linear algebra and algebraic geometry. In classical linear algebra, there exist various criteria to test whether two square matrices commute. Similarly, in the realm of tropical linear algebra, we determine conditions for two tropical matrices that are Kleene stars to commute. Shifting to an algebro-geometric perspective, we explicitly compute the tropicalization of the classical variety of commuting matrices in dimensions 2 and 3. [ABSTRACT FROM AUTHOR]

The action of the mixed complex Monge-Ampère operator ( h, ..., h) ↦ dd h ∧ ... ∧ dd h on piecewise linear functions h is considered. The language of Monge-Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge-Ampère operator depends only on the product of the functions h. [ABSTRACT FROM AUTHOR]

The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety X in a tropicalised toric variety: the normal tropical hypersurface to this Chow polytope is the Minkowski sum of X and a reflected skeleton of the fan of the toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. We show also that Chow polytope subdivisions do not fully distinguish the combinatorial types of tropical variety, and record a proof of the equivalence of two standard definitions of tropical linear spaces. [ABSTRACT FROM AUTHOR]

We study the problem of when the collection of the recession cones of a polyhedral complex also forms a complex. We exhibit an example showing that this is no always the case. We also show that if the support of the given polyhedral complex satisfies a Minkowski-Weyl-type condition, then the answer is positive. As a consequence, we obtain a classification theorem for proper toric schemes over a discrete valuation ring in terms of complete strongly convex rational polyhedral complexes. [ABSTRACT FROM AUTHOR]

We give an algorithm to compute term-by-term multivariate Puiseux series expansions of series arising as local parametrizations of zeroes of systems of algebraic equations at singular points. The algorithm is an extension of Newton's method for plane algebraic curves, replacing the Newton polygon by the tropical variety of the ideal generated by the system. As a corollary we deduce a property of tropical varieties of quasi-ordinary singularities. [ABSTRACT FROM AUTHOR]

Abstract: The tropical variety of a -dimensional prime ideal in a polynomial ring with complex coefficients is a pure -dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gröbner fan software Gfan. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms. [Copyright &y& Elsevier]