MOTIVATION: While there are software packages that analyze Boolean, ternary, or other multi-state models, none compute the complete state space of function-based models over any finite set. Results: We propose Cyclone, a simple light-weight software package which simulates the complete state space for a finite dynamical system over any finite set. AVAILABILITY AND IMPLEMENTATION: Source code is freely available at https://github.com/discretedynamics/cyclone under the Apache-2.0 license.

Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger’s algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger’s algorithm that uses reinforcement learning agents to perform S-pair selection, a key step in the algorithm. We then study how the difficulty of the problem depends on the choices of domain and distribution of polynomials, about which little is known. Finally, we train a policy model using proximal policy optimization (PPO) to learn S-pair selection strategies for random systems of binomial equations. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.

We report on our experiences exploring state of the art Gröbner basis computation. We investigate signature based algorithms in detail. We also introduce new practical data structures and computational techniques for use in both signature based Gröbner basis algorithms and more traditional variations of the classic Buchberger algorithm. Our conclusions are based on experiments using our new freely available open source standalone C++ library.

We give a self-contained exposition of Mayr & Meyer's example of a polynomial ideal exhibiting double exponential degrees for the ideal membership problem, and generalise this example to exhibit minimal syzygies of double exponential degree. This demonstrates the existence of subschemes of projective space of double exponential regularity.