Euler Characteristic of real nondegenerate tropical complete intersections

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 number of unbounded faces and (in case of a curve) the genus. Our point of departure is Vigeland's work who considered the special case $k=n-1$ and where all Newton polytopes are standard simplices. We generalize these results to arbitrary $...

Viro method plays an important role in the study of topology of real algebraic hypersurfaces. The T-primitive hypersurfaces we study here appear as the result of Viro's combinatorial patchworking when one starts with a primitive triangulation. We show that the Euler characteristic of the real part of such a hypersurface of even dimension is equal to the signature of its complex part. We use this result to prove the existence of maximal surfaces in some three-dimensional toric...

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 of surfaces in $\mathbb CP^3$, and has been successively generalized by Bertrand, and by Bihan and ertrand. Our proof, different from the previous approaches, is an application of the motivic nearby fiber of semi-stable degenerations. In p...

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 coordinate projections, other simplest Thom--Boardman and multiple point strata of such projections, generalized Calabi--Yau complete intersections, equaltions of polynomial optimization, hyperplane arrangement complements, and many other intere...

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 differential forms, we also look at symmetric or arbitrary forms instead of the usual alternating ones.

We establish variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of tropical hypersurfaces of toric varieties. It follows from these theorems that the integral tropical homology groups of non-singular tropical hypersurfaces which are compact or contained in $\mathbb{R}^n$ are torsion free. We prove a relationship between the coefficients of the $\chi_y$ genera of complex hypersurfaces in toric varieties and Euler characteristics of t...

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.

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 several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic met...

We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective not-necessarily-normal toric varieties. The second part is a foundational account of tropical intersection theory with proofs of some new theorems relating it to classical intersection theory. Re...

We find an algorithm to compute the quadratic Euler characteristic of a smooth projective complete intersection of hypersurfaces of the same degree. As an example, we compute the quadratic Euler characteristic of a smooth projective complete intersection of two generalized Fermat hypersurfaces. The results presented here also form a chapter in the author's thesis, which was submitted on May 30'th, 2023.