Symbolic integration is a core component of computer algebra systems such as Maple, where the selection of an appropriate integration sub-algorithm has traditionally relied on rule-based heuristics. Because different sub-algorithms are suited to different classes of integrands, accurate algorithm selection significantly affects both the quality and efficiency of the integration result. Prior work has applied LSTM and TreeLSTM models to this task, with TreeLSTM achieving stronger performance. We first reproduce the TreeLSTM experimental framework and then introduce a Transformer model for comparison. Multiple Transformer configurations are evaluated on the original dataset. Results show that one Transformer variant matches the reproduced TreeLSTM in exact optimal matching accuracy but lags behind under 5
GCDHEU is a heuristic algorithm for computing greatest common divisors of multivariate polynomials over the integers and remains available in several computer algebra systems. In this paper, we revisit the GCDHEU algorithm from both theoretical and practical viewpoints. On the theoretical side, we show that the standard correctness argument leaves a gap in the genuinely multivariate setting, and we close this gap by establishing a multivariate Cauchy-type bound. This yields a rigorous proof of correctness for the algorithm. On the practical side, we observe that, for multivariate inputs, the order of variable specialization can substantially affect coefficient growth in the induced integer GCD computations. Motivated by this observation, we propose a simple degree-aware variable-ordering heuristic that helps control such growth. A small modification to SymPy’s implementation of GCDHEU yields an average speedup of about 13.7% on our benchmark set.
Motivated by algorithms manipulating parametric polyhedral sets, we discuss the computation of convex hulls of polyhedra. We extend the algorithm proposed by Fukuda, Liebling and Lütolf, by relaxing its assumptions. In particular, we show how to compute the H-representation (resp. V-representation) of the convex hull (resp. the closure of the convex hull) of polyhedral sets P_1, … , P_k , given by H-representation and such that some P_i ’s may not be of maximum dimension (resp. may not be bounded).
New finite element method (FEM) schemes have been developed for solving self-adjoint boundary value problems (BVPs) for a system of N coupled elliptic equations with second-order mixed partial derivatives (PDEs) on multidimensional domains using symbolic-numerical algorithms implemented in Wolfram Mathematica. Basis functions are constructed using multidimensional Hermite interpolation polynomials (HIPs) on hyperparallelepiped meshes that preserve the continuity of derivatives up to a given order of FEM solutions on the boundaries of adjacent finite elements. The Helmholtz problem for hyperellipsoids mapped in generalized non-orthogonal hyperspherical coordinates to hyperparallelepipeds was chosen as a test example. Benchmark calculations of the lower part of the spectra of BVPs confirm the declared order of accuracy of the finite element method schemes.
The Jacobian of a hyperelliptic curve is the subject of many conjectures and open problems in algebraic geometry and number theory. Efficient Jacobian arithmetic plays a crucial role in the experimental investigation of these problems. For curves of low genus, explicit formulas effectively realize this arithmetic, but manually developing such formulas is not feasible in higher genera. We introduce the first software package for automating the process of generating and verifying explicit formulas for curves of arbitrary genus. Our formulas outperform generic algorithms for Jacobian arithmetic and exhibit comparable operation counts against manually fine-tuned formulas for small genera.
