Learning Fast Monomial Orders for Gröbner Basis Computations

The efficiency of Gröbner basis computation, the standard engine for solving systems of polynomial equations, depends on the choice of monomial ordering. Despite a near-continuum of possible monomial orders, most implementations rely on static heuristics such as GrevLex, guided primarily by expert intuition. We address this gap by casting the selection of monomial orderings as a reinforcement learning problem over the space of admissible orderings. Our approach leverages domain-informed reward signals that accurately reflect the computational cost of Gröbner basis computations and admits efficient Monte Carlo estimation. Experiments on benchmark problems from systems biology and computer vision show that the resulting learned policies consistently outperform standard heuristics, yielding substantial reductions in computational cost. Moreover, we find that these policies resist distillation into simple interpretable models, providing empirical evidence that deep reinforcement learning allows the agents to exploit non-linear geometric structure beyond the scope of traditional heuristics.

Approximate Low-Rank Decomposition for Real Symmetric Tensors

We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition from an algorithmic perspective. More precisely, we prove theorems and design algorithms for the following problem: Suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ error tolerance with respect to $||.||$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We provide two different theoretical bounds and three algorithms for approximate symmetric tensor rank estimation. Our first result is a randomized energy increment algorithm for the case of $L_p$-norms. Our second result is a simple sampling-based algorithm, inspired by some techniques in geometric functional analysis, that works for any norm. We also provide a supplementary algorithm in the case of the Hilbert-Schmidt norm. All our algorithms come with rigorous complexity estimates, which in turn yield our two main theorems on symmetric tensor rank with $\varepsilon$-room of tolerance. We also report on our experiments with a preliminary implementation of the energy increment algorithm.