We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gröbner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. Thus our algorithm has a chance to produce fine decompositions on more structured systems where ensuring genericity assumptions often prohibits exploiting the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.

The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the associative algebra associated to the quantum automorphism group is commutative. We use noncommutative Gröbner bases in order to tackle this problem; the implementation uses Gap and Singular:Letterplace. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

We present a new open source C library msolve dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of msolve relies on Gröbner bases and linear algebra based algorithms for polynomial system solving. It relies on Gröbner basis computation w.r.t. the degree reverse lexicographical order, Gröbner conversion to a lexicographical Gröbner basis and real solving of univariate polynomials. We explain in detail how these three main steps of the solving process are implemented, how we exploit AVX2 instruction processors and the more general implementation ideas we put into practice to better exploit the computational capabilities of this algorithmic framework. We compare the practical performances of msolve with leading computer algebra systems such as Magma, Maple, Singular on a wide range of systems with finitely many complex solutions, showing that msolve can tackle systems which were out of reach by the computer algebra software state-of-the-art.

In this paper we present first steps in using signature-based Gröbner basis algorithms like Faugère's F5 or GVW for computation over Euclidean rings. We present problems appearing when having to deal with coefficients and zero divisors and give practical solution techniques. A hybrid algorithm is presented trying to combine the advantages of signature-based and non-signature-based Gröbner basis computation. For some examples speedups are achieved due to faster finding good reducers with the hybrid technique.

This is a system paper about a new GPLv2 open source C library GBLA implementing and improving the idea [8] of Faugère and Lachartre (GB reduction). We further exploit underlying structures in matrices generated during Gröbner basis computations in algorithms like F4 or F5 taking advantage of block patterns by using a special data structure called multilines. Moreover, we discuss a new order of operations for the reduction process. In various different experimental results we show that GBLA performs better than GB reduction or Magma in sequential computations (up to 40% faster) and scales much better than GB reduction for a higher number of cores: On 32 cores we reach a scaling of up to 26. GBLA is up to 7 times faster than GB reduction. Further, we compare different parallel schedulers GBLA can be used with. We also developed a new advanced storage format that exploits the fact that our matrices are coming from Gröbner basis computations, shrinking storage by a factor of up to 4. A huge database of our matrices is freely avail- able with GBLA.

We introduce the RB algorithm for Gröbner basis computation, a simpler yet equivalent algorithm to F5GEN. RB contains the original unmodified F5 algorithm as a special case, so it is possible to study and understand F5 by considering the simpler RB. We present simple yet complete proofs of this fact and of F5's termination and correctness. RB is parametrized by a rewrite order and it contains many published algorithms as special cases, including SB.

Abstract This paper describes a Buchberger-style algorithm to compute a Grobner basis of a polynomial ideal, allowing for a selection strategy based on signatures. We explain how three recent algorithms can be viewed as different strategies for the new algorithm, and how other selection strategies can be formulated. We describe a fourth as an example. We analyze the strategies both theoretically and empirically, leading to some surprising results.