We present the results of our recent article [4] and discuss its applications [5]. A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the compact torus. We study the structure of the orbit space as the image of the fundamental invariants. For the Weyl groups associated to crystallographic root systems of types A, B, C, D, this image is a compact basic semi-algebraic set. We give the defining polynomial inequalities explicitly as the positivity-locus of a Hermite matrix polynomial. As an application, we consider the problem of computing the optimal value of an exponential function and solve it with algebraic methods under symmetry assumptions.

We aim to factor a sparse polynomial a ∈ Z[ x 1 , ···, x n ] represented by a black box. The authors have previously developed efficient sparse Hensel lifting algorithms for the monic and square-free case that outperforms the algorithm by Kaltofen and Trager in 1990. We complete this black box factorization problem for the non-monic case with a new algorithm that computes the factors of a using many non-monic bivariate Hensel lifts. Our algorithm handles all cases of input a ∈ Z[ x 1 , ···, x n ] including the non-square-free and the non-primitive cases. We have implemented the algorithm in Maple with all major subroutines coded in C for efficiency.

We present Mathematica tools for proving or disproving whether a set of objects constitutes a vector space. When necessary axioms are upheld, the relationships between the variables are presented. When the axioms fail, intuitive counterexamples are shown. A number of different kinds of vectors are demonstrated, with varying types of vector addition and scalar multiplication as well. All of the calculations are performed in an automated fashion.

Each quarter we are pleased to present abstracts of recent doctoral dissertations in Computer Algebra and Symbolic Computation. We encourage all recent Ph.D. graduates who have defended in the past two years (and their supervisors), to submit their abstracts for publication in CCA.

In this work we present an ongoing project on the development and study of new spectral systems which combine filtrations associated to Serre and Eilenberg-Moore spectral sequences of different fibrations. Our new spectral systems are part of a new module for the Kenzo system and can be useful to deduce new relations on the initial spectral sequences and to obtain information about different filtrations of the homology groups of the fiber and the base space of the fibrations.

As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the best performance out of their machine without deep knowledge of code optimization. Naturally, users need different term types either to have different algebraic properties for them, or to use efficient data structures. To this end, we developed Symbolics.jl, an extendable symbolic system which uses dynamic multiple dispatch to change behavior depending on the domain needs. In this work we detail an underlying abstract term interface which allows for speed without sacrificing generality. We show that by formalizing a generic API on actions independent of implementation, we can retroactively add optimized data structures to our system without changing the pre-existing term rewriters. We showcase how this can be used to optimize term construction and give a 113x acceleration on general symbolic transformations. Further, we show that such a generic API allows for complementary term-rewriting implementations. Exploiting this feature, we demonstrate the ability to swap between classical term-rewriting simplifiers and e-graph-based term-rewriting simplifiers. We illustrate how this symbolic system improves numerical computing tasks by showcasing an e-graph ruleset which minimizes the number of CPU cycles during expression evaluation, and demonstrate how it simplifies a real-world reaction-network simulation to halve the runtime. Additionally, we show a reaction-diffusion partial differential equation solver which is able to be automatically converted into symbolic expressions via multiple dispatch tracing, which is subsequently accelerated and parallelized to give a 157x simulation speedup. Together, this presents Symbolics.jl as a next-generation symbolic-numeric computing environment geared towards modeling and simulation.

We introduce a new method for finding a non-realizability certificate of a simplicial sphere Σ. It enables us to prove for the first time the non-realizability of a balanced 2-neighborly 3-sphere by Zheng, a family of highly neighborly centrally symmetric spheres by Novik and Zheng, and several combinatorial prismatoids introduced by Criado and Santos. The method, implemented in the polymake framework, uses integer programming to find a monomial combination of classical 3-term Plücker relations that must be positive in any realization of Σ; but since this combination should also vanish identically, the realization cannot exist. Previous approaches by Firsching, implemented using SCIP, and by Gouveia, Macchia and Wiebe, implemented using Singular and Macaulay2, are not able to process these examples.

We present an Hermitian version of the classical floating-point LLL reduction algorithm of Nguyen and Stehlé[16]. This new variant works on imaginary quadratic fields which are norm-Euclidean and also for some adequate cyclotomic fields. An optimized C++ implementation has been performed, based on the fpLLL code and results show a significant improvement for Hermitian lattices reduction of dimension N when compared to fpLLL reduction on the corresponding Euclidean lattice of dimension 2 N. We demonstrate our implementation in the special case of the Gaussian integers.

We present a SageMath implementation of the symbolic-numeric algorithm introduced by van der Hoeven in 2007 for factoring linear differential operators whose coefficients are rational functions.

The International Conference "Computer Algebra" was held online via Zoom on June 28-29, 2021 and the conference web-site is http://www.ccas.ru/ca/conference. Co-organized by the Dorodnicyn Computing Center of Federal Research Center "Computer Science and Control" of Russian Academy of Science and the Peoples' Friendship University of Russia, the conference was devoted to inspiring discussions on computer algebra and related topics. Researchers from different countries presented talks on their latest research work. This is the fourth edition of the conference, and the previous three were in 2016, 2017 and 2019, respectively.