Related papers: Defect Polytopes and Counter-Examples With polymak…
Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.
Convex polyhedra are the basis for several abstractions used in static analysis and computer-aided verification of complex and sometimes mission critical systems. For such applications, the identification of an appropriate…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by…
We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
In Mathematics is common to make a mistake and therefore a false conclusion arises. In each case it is important to recognize the mistake in order to avoid a similar one in the future. Geometric figures provide decisive help in order to…
We discuss a construction that gives counterexamples to various questions of unique determination of convex bodies.
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…
For certain tropical quartic curves the existing techniques could not predict the lifting behavior of their bitangents over the real numbers. We close this gap by using patchworking techniques. Further, this paper provides an analysis of…
This paper describes a method used to construct infinitely many probable counterexamples of the abc conjecture over the rational integers.
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…
We generate anti-self-polar polytopes via a numerical implementation of the gradient flow induced by the diameter functional on the space of all finite subsets of the sphere, and prove related results on the critical points of the diameter…
In this paper we propose counterexamples to the Geometrization Conjecture and the Elliptization Conjecture.
We introduce tropical analogues of the notion of volume of polytopes, leading to a tropical version of the (discrete) classical isoperimetric inequality. The planar case is elementary, but a higher-dimensional generalization leads to an…
This paper reexamines univariate reduction from a toric geometric point of view. We begin by constructing a binomial variant of the $u$-resultant and then retailor the generalized characteristic polynomial to fully exploit sparsity in the…
Toric geometry provides a bridge between algebraic geometry and combinatorics of fans and polytopes. For each polarized toric variety (X,L) we have associated a polytope P. In this thesis we use this correspondence to study birational…
OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
Computer algebra systems are a great help for mathematical research but sometimes unexpected errors in the software can also badly affect it. As an example, we show how we have detected an error of Mathematica computing determinants of…
The lectures are devoted to a remarkable class of $3$-dimensional polytopes, which are mathematical models of the important object of quantum physics, quantum chemistry and nanotechnology -- fullerenes. The main goal is to show how results…