Predicting the cardinality and maximum degree of a reduced Gröbner basis

Abstract

We construct neural network regression models to predict key metrics of complexity for Gröbner bases of binomial ideals. This work illustrates why predictions with neural networks from Gröbner computations are not a straightforward process. Using two probabilistic models for random binomial ideals, we generate and make available a large data set that is able to capture sufficient variability in Gröbner complexity. We use this data to train neural networks and predict the cardinality of a reduced Gröbner basis and the maximum total degree of its elements. While the cardinality prediction problem is unlike classical problems tackled by machine learning, our simulations show that neural networks, providing performance statistics such as $r^2 = 0.401$, outperform naive guess or multiple regression models with $r^2 = 0.180$.