Related papers: Euler Characteristic of real nondegenerate tropica…

We give a motivic proof of the fact that for non-singular real tropical complete intersections, the Euler characteristic of the real part is equal to the signature of the complex part. This has originally been proved by Itenberg in the case…

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…

We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both…

We study tropical degree bounds, stable tropical intersections, and tropical B\'ezout-type estimates through the geometry of Newton polytopes, mixed subdivisions, and lattice indices. We establish an upper bound for the tropical degree of a…

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…

We prove the structure theorem of the intersection complexes of toric varieties in the category of mixed Hodge modules. This theorem is due to Bernstein, Khovanskii and MacPherson for the underlying complexes with rational coefficients. As…

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…

Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the…

We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…

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…

In this paper we study duality for evaluation codes on intersections of d hypersurfaces with given d-dimensional Newton polytopes, so called toric complete intersection codes. In particular, we give a condition for such a code to be…

We give a definition of Newton non degeneracy independent of the system of generators defining the variety. This definition extends the notion of Newton non degeneracy to varieties that are not necessarily complete intersection. As in the…

This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections,…

We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a…

We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…

We consider the tropicalization of tangent lines to a complete intersection curve $X$ in $\mathbb{P}^n$. Under mild hypotheses, we describe a procedure for computing the tropicalization of the image of the Gauss map of $X$ in terms of the…

A complete intersection $f_1=\cdots=f_k=0$ is sch\"on, if $f_1=\cdots=f_j=0$ defines a sch\"on subvariety of an algebraic torus for every $j\leqslant k$. This class includes nondegenerate complete intersections, critical loci of their…

We discuss conditions for complete intersections in a toric variety which allow to compute Hodge numbers if the complete intersection is a quasi-smooth complete variety. A preliminary step is the computation of the Euler characteristic of…

We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…