Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is unbounded with respect to modality and Milnor number. This method is based on proving that all right-equivalence classes within a mu-constant stratum can be represented by a single normal form derived from a regular basis of a suitably selected special fiber. While both Arnold's and our preceding work on normal forms addresses the determination of a normal form family containing the given germ, this paper takes the next natural step: In arxiv.org/abs/2402.05093 we present an algorithm for computing for a given germ the values of the moduli parameters in its normal form family, that is, a normal form equation in its stable equivalence class. This algorithm will be crucial for understanding the moduli stacks of such singularities. The implementation of this algorithm, along with the foundational classification techniques, is implemented in the library arnold.lib for the computer algebra system Singular.

This paper is joint work with Magdaleen Marais and Gerhard Pfister.

Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift the results to the rationals. In arxiv.org/abs/2401.11606, we present a novel, massively parallel framework for modular computations with polynomial data, which is able to cover a broad spectrum of applications in commutative algebra and algebraic geometry. We demonstrate the framework's effectiveness in Groebner basis computations over the rationals and algorithmic methods from birational geometry. In particular, we develop algorithms to compute images and domains of rational maps, as well as determining invertibility and computing inverses.
Our implementation is based on the Singular/GPI-Space framework, which uses the computer algebra system Singular as computational backend, while coordination and communication of parallel computations is handled by the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language. Convenient installation is realized through the package manager Spack. Relying on Petri nets, our approach provides automated parallelization and balancing of the load between computation, lifting, stabilization testing, and potential verification. We use error tolerant rational reconstruction to ensure termination as long as for a fixed computation there exist only finitely many bad primes. Via stabilization testing, our approach automatically finds with high probablity a minimal set of primes required for the successful reconstruction.
We present timings to illustrate the potential for a game changing improvement of performance over previous modular and non-modular methods. In particular, we illustrate that the approach scales very well with the number of processor cores used for the computation.

This paper is joint work with Dirk Basson, Magdaleen Marais, Mirko Rahn, and Patrick Rakotoarisoa.

In the paper arxiv.org/abs/2404.12085 (in The Computer Algebra System OSCAR) we give illustrative examples of how the computer algebra system OSCAR can support research in commutative algebra and algebraic geometry. We start with a thorough introduction to Groebner basis techniques, with particular emphasis on the computation of syzygies, then apply these techniques to deal with ideal and ring theoretic concepts such as primary decomposition and normalization, and finally use them for geometric case studies which concern curves and surfaces, both from a local and global point of view.

This paper is joint work with Wolfram Decker, and Frank-Olaf Schreyer.

In the paper arxiv.org/abs/2311.11381 (in The Computer Algebra System OSCAR), we present an enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on Gromov-Witten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. We have implemented the algorithm both using Singular and OSCAR. The implementations outperform by far the current method provided in Singular. The OSCAR implementation, benefiting in particular from just-in-time compilation, again by far outperforms the implementation of the new algorithm in Singular. This advancement in computing the Gromov-Witten invariants facilitates a study of number theoretic and geometric properties of the generating series, including quasi-modularity and homogeneity.

This paper is joint work with Alain Hoffmann, Firoozeh Dastur, Hannah Markwig and Ali Traore.

The GIT-fan assigned to an algebraic variety with the action of an algebraic group is a polyhedral fan enumerating all GIT-quotients in the sense of Mumford. In this paper (in The Computer Algebra System OSCAR), we discuss an algorithm to compute the GIT-fan for torus actions on affine varieties with finite symmetries, and its implementation in OSCAR. Relying on Gröbner basis techniques to decide monomial containment, computations with rational polyhedral cones, and expanding orbits with respect to actions of permutation groups, the algorithm combines computational methods from commutative algebra, convex geometry and group theory. Consequently, OSCAR, for the first time, provides an open-source framework that seamlessly manages every stage of the algorithm. We illustrate the algorithm through examples.

This paper is joint work with Thomas Breuer.

In the paper arxiv.org/abs/2305.08783 (Computer Physics Communications Vol. 295, 2024, 108999), we present the package NeatIBP, which automatically generates small-size integration-by-parts (IBP) identities for Feynman integrals. Based on the syzygy and module intersection techniques, the generated IBP identities' propagator degree is controlled and thus the size of the system of IBP identities is shorter than that generated by the standard Laporta algorithm. This package is powered by the computer algebra systems Mathematica and Singular, and the library SpaSM. It is parallelized on the level of Feynman integral sectors. The generated small-size IBP identities can subsequently be used for either finite field reduction or analytic reduction. We demonstrate the capabilities of this package on several multi-loop IBP examples.

This paper is joint work with Zihao Wu, Rourou Ma, Hefeng Xu, and Yang Zhang.

In the paper arxiv.org/abs/2202.02986 (Journal of High Energy Physics 05 (2022) 68, 66 pp) we uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solutions of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of the Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the, closely related, random-cluster Potts model on medium-size lattices with toroidal boundary conditions by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

This paper is joint work with Jesper Lykke Jacobsen, Yunfeng Jiang, and Yang Zhang.

In the ISSAC 2021 Tutorial Massively Parallel Computations in Algebraic Geometry (Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation (2021), 11-14), we give an introduction to the Singular/GPI-Space framework for massively parallel computations in computer algebra. Being based on the idea of separation of computation and coordination, this framework relies on the Petri net based workflow management system GPI-Space for the coordination layer, while the computer algebra system Singular is used as the frontend and computational backend. We illustrate the use of this approach in examples taken from the context of algebraic geometry.

This paper is joint work with Anne Frühbis-Krüger.

Multivariate partial fractioning is a powerful tool for simplifying rational function coefficients in scattering amplitude computations. Since current research problems lead to large sets of complicated rational functions, performance of the partial fractioning as well as size of the obtained expressions are a prime concern. In arxiv.org/abs/2104.06866 (Computer Physics Communications Vol. 294, 2024, 108942), we develop a large scale parallel framework pfd-parallel for multivariate partial fractioning, which implements and combines an improved version of Leinartas' algorithm and the MultivariateApart algorithm. Our approach relies only on open source software. It combines parallelism over the different rational function coefficients with parallelism for individual expressions. The implementation is based on the Singular/GPI-Space framework for massively parallel computer algebra, which formulates parallel algorithms in terms of Petri nets. The modular nature of this approach allows for easy incorporation of future algorithmic developments into our package. We demonstrate the performance of our framework by simplifying expressions arising from current multiloop scattering amplitude problems. We discover that the compression ratio increases with the complexity of the Feynman integrals. See here for the analytic result.

This paper is joint work with Dominik Bendle, Murray Heymann, Mirko Rahn, Lukas Ristau, Marcel Wittmann, Zihao Wu, Yingxuan Xu and Yang Zhang.

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 paper arxiv.org/abs/2010.10185, 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.

This paper is joint work with Magdaleen Marais and Gerhard Pfister.

In the manuscript arxiv.org/abs/2010.06895 (Proceedings of the Conference "MathemAmplitudes 2019" in Padova, Italy, PoS - Proceedings of Science MA2019 004 (2022), 19pp) we provide an overview of the module intersection method for the the integration-by-parts (IBP) reduction of multi-loop Feynman integrals. The module intersection method, based on computational algebraic geometry, is a highly efficient way of getting IBP relations without double propagator or with a bound on the highest propagator degree. In this manner, trimmed IBP systems which are much shorter than the traditional ones can be obtained. We apply the modern, Petri net based, workflow management system GPI-Space in combination with the computer algebra system Singular to solve the trimmed IBP system via interpolation and efficient parallelization. We show, in particular, how to use the new plugin feature of GPI-Space to manage a global state of the computation and to efficiently handle mutable data. Moreover, a Mathematica interface to generate IBPs with restricted propagator degree, which is based on module intersection, is presented in this review.

This paper is joint work with Dominik Bendle, Wolfram Decker, Alessandro Georgoudis, Franz-Josef Pfreundt, Mirko Rahn, and Yang Zhang.

In the paper arxiv.org/abs/2008.13194 (Journal of High Energy Physics (2020) 54, 39 pp) we present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas' multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, We observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ~100. We observe that our algorithm also works well for settings without a UT basis.

This paper is joint work with Marcel Wittmann, Zihao Wu, Yingxuan Xu, and Yang Zhang.

In the paper arxiv.org/abs/2003.13752 we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr0(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).

This paper is joint work with Dominik Bendle, Yue Ren, and Benjamin Schröter.

In the paper arxiv.org/abs/1908.09575 the behavior of the randomized breadth-first search algorithm is analyzed on arbitrary regular and non-regular graphs. Our argument is based on the expander mixing lemma, which entails that the results are strongest for Ramanujan graphs, which asymptotically maximize the spectral gap. We compare our theoretical results with computational experiments on flip graphs of point configurations. The latter is relevant for enumerating triangulations.

This paper is joint work with Michael Joswig, Lars Kastner, and Andrew Newman.

In the paper arxiv.org/abs/1908.04301 (Journal of High Energy Physics 02 (2020) 79, 34 pp) we introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which is being developed at the Fraunhofer Institute for Industrial Mathematics (ITWM). In our approach, the IBP relations are first trimmed by modern algebraic geometry tools and then solved by sparse linear algebra and our new interpolation methods. These steps are efficiently automatized and automatically parallelized by modeling the algorithm in GPI-Space using the language of Petri-nets. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.

This paper is joint work with Dominik Bendle, Wolfram Decker, Alessandro Georgoudis, Franz-Josef Pfreundt, Mirko Rahn, Pascal Wasser, and Yang Zhang.

In the paper arxiv.org/abs/1812.04936 (Annales de l’Institut Henri Poincaré D, Comb. Phys. Interact. 9 (2022), 121-158) we study generating series of Gromov-Witten invariants of the product of an elliptic curve E with projective 1-space and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are - just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals - complex analytic path integrals involving a product of propagators (equal to the Weierstrass-℘-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in a tropical sum of tropical Hirzebruch surfaces.

This paper is joint work with Christoph Goldner and Hannah Markwig.

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 the note arxiv.org/abs/1811.06092 (Computeralgebrarundbrief 64 (2019) 8-13), 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.

This paper is joint work with Anne Frühbis-Krüger and Mirko Rahn.

In the paper arxiv.org/abs/1809.10659 (SIGMA 18 (2022) 046, 30 pp) we prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing a tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve. For the case of the elliptic curve, the tropical version of mirror symmetry holds on a fine level and easily implies the equality of the generating series of descendant Gromov-Witten invariants of the elliptic curve to Feynman integrals. To prove tropical mirror symmetry for elliptic curves, we investigate the bijection between graph covers and sets of monomials contributing to a coefficient in a Feynman integral. We also soup up the traditional approach in mathematical physics to mirror symmetry for the elliptic curve, involving operators on a Fock space, to give a proof of tropical mirror symmetry for Hurwitz numbers of the elliptic curve. In this way, we shed light on the intimate relation between the operator approach on a bosonic Fock space and the tropical approach.

This paper is joint work with Christoph Goldner and Hannah Markwig.

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high performance numerical simulation, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In the paper arxiv.org/abs/1808.09727 (Foundations of Computational Mathematics (2020)) we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematical core algorithms are implemented in the computer algebra system Singular, whose focus is on polynomial computations, while coordination is handled by the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language, and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka's celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores.

This paper is joint work with Wolfram Decker, Anne Frühbis-Krüger, Franz-Josef Pfreundt, Mirko Rahn, and Lukas Ristau.

In the paper arxiv.org/abs/1805.01873 (Journal of High Energy Physics 09 (2018) 24, 30 pp) we present the powerful module-intersection integration-by-parts (IBP) method, suitable for multi-loop and multi-scale Feynman integral reduction. Utilizing modern computational algebraic geometry techniques, this new method successfully trims traditional IBP systems dramatically to much simpler integral-relation systems on unitarity cuts. We demonstrate the power of this method by explicitly carrying out the complete analytic reduction of two-loop five-point non-planar hexagon-box integrals, with degree-four numerators, to a basis of 73 master integrals.

This paper is joint work with Alessandro Georgoudis, Kasper J. Larsen, Hans Schönemann, and Yang Zhang.

In this paper arxiv.org/abs/1811.06093 (German, Beiträge zum Mathematikunterricht, Proceedings of the joint meeting of GDM and DMV, Paderborn 2018, 325-328) 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 practial programming projects and its role as a research topic in student papers.

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. In the paper arxiv.org/abs/1712.09737 (Phys. Rev. D 98, 2018, 025023) we present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

This paper is joint work with Alessandro Georgoudis, Kasper J. Larsen, Mathias Schulze, Yang Zhang.

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 the paper arxiv.org/abs/1702.06920 for ICMS 2016 (Lecture Notes in Computer Science 9725 (2016), 93-102), we illustrate 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 explicit 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.

This paper is joint work with Wolfram Decker, Claus Fieker, Santiago Laplagne and Gerhard Pfister.

V.I. Arnold has described 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 the paper arxiv.org/abs/1604.04774 (Singularities and Computer Algebra - Festschrift for Gert-Martin Greuel on the Occasion of his 70th Birthday, Springer 2017, 21-46), 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. We have implemented our algorithm in the library classify2.lib for the computer algebra system Singular.

This paper is joint work with Magdaleen Marais and Gerhard Pfister.

In the paper arxiv.org/abs/1603.09241 (Mathematics of Computation (2020)) we present an algorithm to compute the GIT-fan for torus actions on affine varieties with symmetries. The algorithm combines computational techniques from commutative algebra, convex geometry and group theory. We have implemented our algorithm in the library gitfan.lib for the computer algebra system Singular. Using our implementation, we compute the Mori chamber decomposition of the cone of movable divisors of the Deligne-Mumford moduli space of 6-pointed stable curves of genus 0. The resulting GIT-fan data can be read out from a list of hashes representing cones through the library M06chambers.lib.

This paper is joint work with Simon Keicher and Yue Ren.

Based on an idea in Hironaka's proof of resolution of singularities, we present in arxiv.org/abs/1602.04522 (J. Symb. Comp. 86 (2017), 153-165) an algorithmic smoothness test for algebraic varieties. The test is inherently parallel and does not involve the calculation of codimension-sized minors of the Jacobian matrix of the variety. We also describe a hybrid method which combines the new method with the Jacobian criterion, thus making use of the strengths of both approaches. We have implemented all algorithms in the library smoothtst.lib for the computer algebra system Singular, and compare the different approaches with respect to timings and memory usage. The test examples originate from questions in algebraic geometry, where the use of the Jacobian criterion is impractical due to the number and size of the minors involved.

This paper is joint work with Anne Frühbis-Krüger.

In arxiv.org/abs/1512.09028 (J. Symb. Comp. 99 (2020), 250-282) we present a classification algorithm for isolated hypersurface singularities of corank 2 and modality 1 over the real numbers. For a singularity given by a polynomial over the rationals, the algorithm determines its right equivalence class by specifying all representatives in Arnold's list of normal forms (Arnold et al. 1985) belonging to this class, and the corresponding values of the moduli parameter. We discuss how to computationally realize the individual steps of the algorithm for all singularities in consideration, and give explicit examples. The algorithm is implemented in the Singular library realclassify.lib.

The paper is joint work with Magdaleen Marais and Andreas Steenpaß.

In arxiv.org/abs/1505.05054 (J. Symb. Comp. (2020)) we present a new algorithm for computing integral bases in algebraic function fields, or equivalently for constructing the normalization of a plane curve. Our basic strategy makes use of localization and, then, completion at each singularity of the curve. In this way, we are reduced to finding integral bases at the branches of the singularities. To solve the latter task, we work with suitably truncated Puiseux expansions. In contrast to van Hoeij's algorithm, which also relies on Puiseux expansions (but pursues a different strategy), we use Hensel's lemma as a key ingredient. This allows us to compute factors corresponding to groups of conjugate Puiseux expansions, without actually computing the individual expansions. In this way, we make substantially less use of the Newton-Puiseux algorithm. In addition, our algorithm is inherently parallel. As a result, it outperforms in most cases any other algorithm known to us by far. Typical applications are the computation of adjoint ideals and, based on this, the computation of Riemann-Roch spaces and the parametrization of rational curves.

The paper is joint work with Wolfram Decker, Santiago Laplagne and Gerhard Pfister.

In arxiv.org/abs/1505.05040 (in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Springer 2017) 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.

The paper is joint work with Wolfram Decker, Santiago Laplagne and Gerhard Pfister.

An important open question in algebraic combinatorics is whether for a simplicial sphere, or more generally for a Gorenstein* simplicial complex, the f-vector satisfies McMullen's g-conjecture. Assume σ is a face of a Gorenstein* simplicial complex D. For the g-conjecture to hold it is enough to prove that the Stanley-Reisner ring of D over the real numbers satisfies the Weak Lefschetz Property. In arxiv.org/abs/1501.01513 (Australas. J. Combin. 76 (2020), 266-287) 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_σ] of the stellar subdivision D_σ of D at σ. We prove that this is the case if 2(dim σ) > dim(D)+1. If this equivalence was to be proven without any assumptions on the dimension of σ, it would then have as a corollary the g-conjecture for the class of PL-spheres. In the last section we give some further results and constructions which could, perhaps, prove useful in attacking the problem in general.

The paper is joint work with Stavros Papadakis.

3D printing and other layer manufacturing processes are challenged by dimensional accuracy. In http://arxiv.org/abs/1512.09036 (ICMS 2016, Lecture Notes in Computer Science 9725 (2016), 117-126) we propose to use objects from algebraic geometry as test shapes. Processes with the risk of deformation after time or post processing may find this technique beneficial. 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. The cubic offers simplicity to the dimensional comparison process, in that the lines and many other features can be analytically determined and easily measured. For example, the surface contains so-called Eckhard points, in each of which three of the lines intersect. 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.

The paper is joint work with Magdaleen Marais and Andre van der Merwe.

Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. In arxiv.org/abs/1309.5893 (J. Reine Angew. Math. (Crelles Journal), Vol. 2017, Issue 732, 211–246 DOI: 10.1515/crelle-2014-0143) we prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov-Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

The paper is joint work with Kathrin Bringmann, Arne Buchholz and Hannah Markwig.

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert-Remmert-type algorithms rely on so-called test ideals.
For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In http://arxiv.org/abs/1309.4620 (IJAC 24, No. 1 (2014), 69-94), we relate the number of iterations for global Grauert-Remmert-type normalization algorithms to that of its local descendants.
We complement our results by an explicit study of ADE singularities. This includes the description of the normalization process in terms of value semigroups of curves. The intermediate steps produce ADE singularities and simple space curve singularities from the list of Fruehbis-Krueger.

The paper is joint work with Wolfram Decker and Mathias Schulze.

In http://arxiv.org/abs/1212.4358 (Hokkaido Math. J. 44 (2015), 341-364) we give a bound for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex by applying unprojection theory. From this we derive a bound for the Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. The bound depends only on the number of subdivisions, and we construct examples which prove that it is sharp (for an implementation of the construction see BettiBounds.m2).

The paper and package are joint work with Stavros Papadakis.

A standard method for computing a rational number from its values modulo a collection of primes is to determine its value modulo the product of the primes via Chinese Remaindering, and then use Farey sequences for rational reconstruction. Successively enlarging the set of primes if needed, this method is guaranteed to work if we restrict ourselves to "good" primes. Depending on the particular application, however, there is often no efficient way of finding good primes. This note shows that in most situations, we can simply ignore this problem. In fact, we present an error tolerant algorithm for rational reconstruction. With regard to applications, we are particularly interested in the design of modular and, thus, parallel versions of algorithms in commutative algebra and algebraic geometry. Here, typically, the final result consists of one or several a priori unknown ideals which are found via constructions yielding the (reduced) Groebner bases of the ideals.

The paper http://arxiv.org/abs/1207.1651 (Mathematics of Computation 84 (2015), 3013-3027) is joint work with Wolfram Decker, Claus Fieker and Gerhard Pfister.

Considering finite extensions of positive affine semigroup rings K[B] of K[A] over a field K we have developed in [BEN] an algorithm to decompose K[B] as a direct sum of monomial ideals in K[A]. By computing the regularity of homogeneous semigroup rings from the decomposition we have confirmed the Eisenbud-Goto conjecture in a range of new cases not tractable by standard methods. Here we first illustrate this technique and its implementation in our Macaulay2 package MonomialAlgebras by computing the decomposition and the regularity step by step for an explicit example. We then focus on ring-theoretic properties of simplicial semigroup rings. From the characterizations given in [BEN] we develop and prove explicit algorithms testing properties like Buchsbaum, Cohen-Macaulay, Gorenstein, normal, and seminormal, all of which imply the Eisenbud-Goto conjecture. All algorithms are implemented in our Macaulay2 package.

The paper http://arxiv.org/abs/1206.1735 (JSAG 5 (2013), 8-14) and the package for Macaulay2 are joint work with David Eisenbud and Max Nitsche.

The paper http://arxiv.org/abs/1110.4299 (J. Symb. Comp. 51 (2013), 99-114) and the Singular packages locnormal.lib and modnormal.lib are joint work with Wolfram Decker, Santiago Laplagne, Gerhard Pfister, Andreas Steenpaß, and Stefan Steidel.

Let A contained in B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular we obtain a fast algorithm to compute the Castelnuovo-Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud-Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.

The paper http://arxiv.org/abs/1110.3653 (Exper. Math. 21(4) (2012), 385-394) and the Macaulay2 package are joint work with David Eisenbud and Max Nitsche.

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. In the joint work http://arxiv.org/abs/1103.2314 (JSAG 4 (2012), 6-12) with Stavros Papadakis we describe an algorithm computing the Kustin-Miller complex and its implementation in the Macaulay2 package KustinMiller. We explain how it can be applied to explicit examples from geometry and commutative algebra.

In the joint work http://arxiv.org/abs/0912.2152 (Osaka J. Math. 49-1 (2012), 81-100) with Stavros Papadakis we study the structure of Stanley-Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex Δ(d;m) of the d-dimensional cyclic polytope with m vertices. We show how to express the Stanley-Reisner ring of Δ(d;m+1) in terms of the Stanley{Reisner rings of Δ(d;m) and Δ(d-2;m-1). As an application, we use the Kustin-Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.

An implementation of this algorithm for the computer algebra system Macaulay2 is available in the package CyclicPolytopeRes.m2, which computes the minimal resolution of the Stanley-Reisner ring of
Δ(d,m) using unprojection theory. It also allows the user to compute the flat unprojection family and output the unprojection structure of Δ(d,m).

In the joint work http://arxiv.org/abs/0912.2152 with Stavros Papadakis (Australas. J. Combin. 55 (2013), 235-247) we relate the theory of unprojection (so in some sense birational geometry) to the theroy of Stanley-Reisner rings. The aim of unprojection theory is 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 flat unprojection families of type Kustin-Miller. As an application, we inductively calculate the minimal graded free resolutions of Stanley-Reisner rings associated to stacked polytopes.

This is an extensive project in progress to provide in the computer algebra system Macaulay2 a computational framework for tropical mirror symmetry and as part of this also deformations of Stanley-Reisner rings. It has efficient data types to store and dualize tropical varieties, polytopes and faces thereof.
You can find the latest stable version of this package also in the Macaulay2 distribution.
This package can use (for performance reasons) the packages ConvexInterface.m2 and MapleInterface.m2. You can also find the latest stable version of this package in the Macaulay2 distribution. I also have a Singular project going on about this topic.

Via tropical geometry, toric geometry, Gröbner bases and deformation theory a general framework for mirror symmetry of Calabi-Yau varieties leading to an algorithmic mirror construction is developed in my PhD thesis. The tropical mirror construction comes with a natural mirror map identifying deformations and divisors. It reproduces the construction by Batyrev for hypersurfaces, the generalization thereof by Batyrev and Borisov to complete intersections and a construction by Roedland for a particular non-complete intersection Pfaffian Calabi-Yau 3-fold. We also obtain a new mirror by applying the construction to the non-complete intersection Pfaffian Calabi-Yau 3-fold of degree 13 in projective 6-space.
Here is a preprint version of my PhD thesis for download (3.8 MB, 544pp): arXiv:0708.4402v1[math.AG].

You will find my implementation of the tropical mirror construction here once it is cleaned up and commented. The implementation is provided as a Maple package interfacing to M2 or Singular for standard basis computations.

As an alternative, there is now a Macaulay2 package providing not all but a good part of the functionality of the Maple package.

Consider a rational curve C in projective 2-space with rational coefficients. If the degree of C is odd then there is a parametrization of C with rational coefficients. If the degree is even then there is a parametrization with coefficients in the rational numbers or a field extension thereof of degree 2.
In my diploma thesis (940KB, 202pp) I give an algorithm computing for any given rational curve such parametrization. The algorithm works without inducing random choices, so leads to small heights of the coefficients and fast computation times.
We compute a birational map of C to a rational normal curve, which then is parametrized. This is based on an idea going back to Hilbert.
An implementation of this algorithm inside the computer algebra system Macaulay2 is available in the package Parametrization.m2, which is computing the rational parametrization of C over projective 1-space if the degree is odd or if the degree is even and the curve has a rational point, or otherwise over a plane conic.

The package Parametrization.m2 requires the AdjointIdeal.m2 package.

A new version of this algorithm has been implemented in the Singular library paraplanecurves.lib. This is joint work with Wolfram Decker, Santiago Laplagne, and Frank Seelisch.

The package AdjointIdeal.m2 for the computer algebra system Macaulay2 computes the geometric genus and the adjoint ideal of a plane curve.

A new version of this algorithm and an entirely new, much faster algorithm have been implemented in the Singular library paraplanecurves.lib. This is joint work with Wolfram Decker, Santiago Laplagne, and Frank Seelisch. A paper on this and a much more powerful algorithm and implementation will be available soon.

For example the computaton of the adjoint ideal in paraplanecurves.lib requires the computation of an integral basis of the algebraic function field. In the Macaulay2 implementation this was done by calling van Hoeij's algorithm.

For the singular implementation a new, much faster algorithm has been developed and implemented in the Singular library integralbasis.lib using localization techniques. This is joint work with Wolfram Decker, Santiago Laplagne, and Frank Seelisch. A paper on this will be available soon.