Related papers: Amoebas of half-dimensional varieties

We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…

We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total…

We show that infinitesimal Torelli for $n$-forms holds for abelian covers of algebraic varieties of dimension $n\ge 2$, under some explicit ampleness assumptions on the building data of the cover. Moreover, we prove a variational Torelli…

This paper is a report based on the results obtained during a three months internship at the University of Pittsburgh by the first author and under the mentorship of the second author. The notion of an amoeba of a subvariety in a torus…

Let $Q_n$ be the cube of side length one centered at the origin in $\mathbb{R}^n$, and let $F$ be an affine $(n-d)$-dimensional subspace of $\mathbb{R}^n$ having distance to the origin less than or equal to $\frac 1 2$, where $0<d<n$. We…

To every finite-dimensional $\mathbb C$-algebra $\Lambda$ of finite representation type we associate an affine variety. These varieties are a large generalization of the varieties defined by "$u$ variables" satisfying "$u$-equations", first…

Answering a question by Nisse and Sottile, we derive a formula for the dimension of the amoeba of an irreducible algebraic variety.

Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.

In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-dimensional spherical polyhedron, then each irreducible component of the…

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in $\mathbb C^n$. We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree…

In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with…

Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [BucherBurgerIozzi2013] we show that the volume of a representation of the…

In the area of symbolic-numerical computation within computer algebra, an interesting question is how "close" a random input is to the "critical" ones, like the singular matrices in linear algebra or the polynomials with multiple roots for…

Global amoebas are a wide and rich family of graphs that emerged from the study of certain Ramsey-Tur\'an problems in $2$-colorings of the edges of the complete graph $K_n$ that deal with the appearance of unavoidable patterns once a…

We consider a compact hyperbolic antiprism. It is a convex polyhedron with $2n$ vertices in the hyperbolic space $\mathbb{H}^3$. This polyhedron has a symmetry group $S_{2n}$ generated by a mirror-rotational symmetry of order $2n$, i.e.…

In this paper we derive an explicit lower bound on the volume of a hyperbolic $n$-orbifold for dimensions greater than or equal to four. Our main tool is H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a…

We present algorithms for computation and visualization of amoebas, their contours, compactified amoebas and sections of three-dimensional amoebas by two-dimensional planes. We also provide method and an algorithm for the computation…

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study…

We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of…

Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for…