Santosh Gnawali

• This thesis explores massively parallel methods for algebraic computations, focusing on parallel algorithms for computing syzygies and free resolutions of finitely generated modules over polynomial rings, relying on ideas from Schreyer's method.Further topics are modular methods for free resolutions and higher syzygy modules over polynomial rings, with applications to computing rational parametrizations of rational normal curves, as well as plane (singular) rational curves. Massively parallel computations are a transformative approach in the field of computer algebra, enabling the handling of complex and large-scale algebraic problems, with applications on the geometric side, that were previously infeasible. In this thesis, we focus on Schreyer's resolution algorithm and demonstrate how this algorithm can be restructured to accommodate parallel computations by modeling the algorithm in the form of a Petri net. This facilitates the coordination of tasks and data structures within a parallel computing environment and allows us to exploit the inherent parallelism of the algorithm. In order to formulate the algorithm as a Petri net, we in particular restructure it by translating the recursive algorithm for computing the tail terms of syzygies into an iterative form. Moreover, we develop a parallel approach for summing up the syzygy terms. For our implementations, we utilize the SINGULAR/GPI-SPACE framework, which combines the computer algebra system SINGULAR [DGPS24] with the workflow management system GPI-SPACE [Fra25], which natively supports an enriched variant of Petri nets as a coordination language for parallel computations. We observe that in our setting with a large number of small computations, executing SINGULAR interpreter code in the computational backend becomes a serious bottleneck. We thus extend the capabilities of the SINGULAR/GPI-SPACE framework to run kernel level computations based on the C++ library version of SINGULAR, and implement our computational building blocks in this way. We expect that this technique will have numerous further applications in the future. A key challenge in algebraic computations over the rational numbers, such as those relying on Buchberger’s algorithm for computing Gröbner bases, is the issue of intermediate coefficient growth [Arn03]. This phenomenon can significantly affect the efficiency and feasibility of computations. Modular and p-adic methods have proven to be successful in addressing intermediate coefficient growth in various applications, see, for example, [Arn03],[IPS11], [BDFP15]. Building on generic massively parallel modular methods, we present a novel approach to parametrizing rational normal curves using massively parallel modular methods. We implement a modular technique for determining free resolutions as well as specific higher syzygy matrices to avoid the computational cost associated with lifting all syzygy matrices. For example, the last syzygy matrix can be used to compute certain Ext modules, which in turn allow one to iteratively find a parametrization of a rational normal curve. This, in particular, illustrates how classical algebraic geometry techniques can be transferred to a parallel context.