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In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the exception of $23$ special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr$^+(3,8)$ and compare them to the cluster complex of the classical Grassmannian Gr$(3,8)$., 32 pages, 9 figures

In his groundbreaking work on classification of singularities with regard to right and stable equivalence of germs, Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with either modality less than or equal to two or Milnor number less than or equal to 16. Moreover, he has described an algorithmic classifier, which determines the type of a given such singularity. In the present paper, we extend Arnold's work to a large class of singularities which is unbounded with regard to modality and Milnor number. We develop an algorithmic classifier, which determines a normal form for any singularity with corank less than or equal to two which is equivalent to a germ with non-degenerate Newton boundary in the sense of Kouchnirenko. In order to realize the classifier, we prove a normal form theorem: Suppose K is a mu-constant stratum of the jet space which contains a germ with a non-degenerate Newton boundary. We first observe that all germs in K are equivalent to some germ with the same fixed non-degenerate Newton boundary. We then prove that all right-equivalence classes of germs in K can be covered by a single normal form obtained from a regular basis of an appropriately chosen special fiber. All algorithms are implemented in the library arnold.lib for the computer algebra system Singular., Comment: 32 pages

In this note for the joint meeting of DMV and GDM we illustrate with examples the role of computer algebra in university mathematics education. We discuss its potential in teaching algebra, but also computer algebra as a subject in its own right, its value in the context of practical programming projects and its role as a research topic in student papers., Comment: German, 5 pages

The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively parallel computations in commutative algebra and algebraic geometry. In this note, we give an overview on the current capabilities of this framework by looking into three sample applications: determining smoothness of algebraic varieties, computing GIT-fans in geometric invariant theory, and determining tropicalizations. These applications employ algorithmic methods originating from commutative algebra, sheaf structures on manifolds, local geometry, convex geometry, group theory, and combinatorics, illustrating the potential of the framework in further problems in computer algebra., Comment: 9 pages, 6 figures

Computations over the rational numbers often suffer from intermediate coefficient swell. One solution to this problem is to apply the given algorithm modulo a number of primes and then lift the modular results to the rationals. This method is guaranteed to work if we use a sufficiently large set of good primes. In many applications, however, there is no efficient way of excluding bad primes. In this note, we describe a technique for rational reconstruction which will nevertheless return the correct result, provided the number of good primes in the selected set of primes is large enough. We give a number of illustrating examples which are implemented using the computer algebra system Singular and the programming language Julia. We discuss applications of our technique in computational algebraic geometry., Comment: 8 pages, 1 figure, 1 table

In (Arnold, 1985), V.I. Arnold has obtained normal forms and has developed a classifier for, in particular, all isolated hypersurface singularities over the complex numbers up to modality 2. Building on a series of 105 theorems, this classifier determines the type of the given singularity. However, for positive modality, this does not fix the right equivalence class of the singularity, since the values of the moduli parameters are not specified. In this paper, we present a simple classification algorithm for isolated hypersurface singularities of corank and modality up to two. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying a polynomial representative in Arnold's list of normal forms., Comment: 19 pages, 5 figures, minor revisions

3D printing and other layer manufacturing processes are challenged by dimensional accuracy. Several techniques are used to validate and calibrate dimensional accuracy through the complete building envelope. The validation process involves the growing and measuring of a shape with known parameters. The measured result is compared with the intended digital model. Processes with the risk of deformation after time or post processing may find this technique beneficial. We propose to use objects from algebraic geometry as test shapes. A cubic surface is given as the zero set of a 3rd degree polynomial with 3 variables. A class of cubics in real 3D space contains exactly 27 real lines. We provide a library for the computer algebra system Singular which, from 6 given points in the plane, constructs a cubic and the lines on it. A surface shape derived from a cubic offers simplicity to the dimensional comparison process, in that the straight lines and many other features can be analytically determined and easily measured using non-digital equipment. For example, the surface contains so-called Eckardt points, in each of which three of the lines intersect, and also other intersection points of pairs of lines. Distances between these intersection points can easily be measured, since the points are connected by straight lines. At all intersection points of lines, angles can be verified. Hence, many features distributed over the build volume are known analytically, and can be used for the validation process. Due to the thin shape geometry the material required to produce an algebraic surface is minimal. This paper is the first in a series that proposes the process chain to first define a cubic with a configuration of lines in a given print volume and then to develop the point cloud for the final manufacturing. Simple measuring techniques are recommended., 8 pages, 1 figure, 1 table

Assume sigma is a face of a Gorenstein* simplicial complex D. We investigate the question of whether the Weak Lefschetz Property of the Stanley-Reisner ring k[D] (over an infinite field k) is equivalent to the same property of the Stanley-Reisner ring k[D_sigma] of the stellar subdivision D_sigma. We prove that this is the case if the dimension of sigma is big compared to the codimension., Comment: 25 pages, 1 figure

We present new algorithms for computing adjoint ideals of curves and thus, in the planar case, adjoint curves. With regard to terminology, we follow Gorenstein who states the adjoint condition in terms of conductors. Our main algorithm yields the Gorenstein adjoint ideal G of a given curve as the intersection of what we call local Gorenstein adjoint ideals. Since the respective local computations do not depend on each other, our approach is inherently parallel. Over the rationals, further parallelization is achieved by a modular version of the algorithm which first computes a number of the characteristic p counterparts of G and then lifts these to characteristic zero. As a key ingredient, we establish an efficient criterion to verify the correctness of the lift. Well-known applications are the computation of Riemann-Roch spaces, the construction of points in moduli spaces, and the parametrization of rational curves. We have implemented different variants of our algorithms together with Mnuk's approach in the computer algebra system Singular and give timings to compare the performance of the algorithms., Comment: 32 pages

Applying tropical geometry a framework for mirror symmetry, including a mirror construction for Calabi-Yau varieties, was proposed by the author. We discuss the conceptual foundations of this construction based on a natural mirror map identifying deformations and divisors. We show how the construction specializes to that by Batyrev for hypersurfaces and its generalization by Batyrev and Borisov to complete intersections. Based on an explicit example we comment on the implementation in the Macaulay2 package SRdeformations., 15 pages, no figures

The Kustin-Miller complex construction, due to A. Kustin and M. Miller, can be applied to a pair of resolutions of Gorenstein rings with certain properties to obtain a new Gorenstein ring and a resolution of it. It gives a tool to construct and analyze Gorenstein rings of high codimension. We describe the Kustin-Miller complex and its implementation in the Macaulay2 package KustinMiller, and explain how it can be applied to explicit examples., Comment: 5 pages, no figures. Package may be downloaded at http://www.math.uni-sb.de/ag/schreyer/jb/Macaulay2/KustinMiller/html/

Unprojection theory aims to analyze and construct complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of non-acyclic Gorenstein simplicial complexes correspond to unprojections of type Kustin-Miller. As an application, we inductively calculate the minimal graded free resolutions of Stanley-Reisner rings associated to stacked polytopes, recovering results of Terai, Hibi, Herzog and Li Marzi., Comment: v2: minor improvements, 14 pages, 1 figure

Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rodland for the degree 14 Calabi-Yau threefold in PP^6 defined by the Pfaffians of a general linear 7x7 skew-symmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Groebner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples., 540 pages, 46 figures