Related papers: Degree Bounds for Amoeba Contours
The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics…
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…
An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log…
We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$ is a very general hypersurface of degree $d\geq 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of…
This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we call toric degenerations. For these, the…
We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into…
The amoeba of a Laurent polynomial $f \in \C[z_1^{\pm 1},\ldots,z_n^{\pm 1}]$ is the image of its zero set $\mathcal{V}(f)$ under the log-absolute-value map. Understanding the space of amoebas (i.e., the decomposition of the space of all…
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…
Exploiting a connection between amoebas and tropical curves, we devise a method for computing tropical curves using numerical algebraic geometry and give an implementation. As an application, we use this technique to compute Newton polygons…
We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…
We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-R\'enyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous [AOP 1997]. In this model, the…
A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. We describe the structure of the boundary of the coamoeba of a variety, which we relate to its logarithmic limit set. Detailed examples of…
We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety…
The algebraic approach to the Constraint Satisfaction Problem (CSP) uses high order symmetries of relational structures -- polymorphisms -- to study the complexity of the CSP. In this paper we further develop one of the methods the…
We consider Gromov's homological higher convexity for complements of tropical varieties, establishing it for complements of tropical hypersurfaces and curves, and for nonarchimedean amoebas of varieties that are complete intersections over…
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…
We study the space of loops into a hypersurface complement, and show that the corresponding topological algebra of Laurent series with coefficients in $\mathcal{O}(L\mathbf{A}^{d}_{f})$ is a topological localisation of…
We study the growth of degrees in many autonomous and non-autonomous lattice equations defined by quad rules with corner boundary values, some of which are known to be integrable by other characterisations. Subject to an enabling…
This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I"…
Various limit-free formulas are given for the computation of the algebraic and the topological entropy, respectively in the settings of endomorphisms of locally finite discrete groups and of continuous endomorphisms of totally disconnected…