Computing the canonical representation of constructible sets

Constructible sets are needed in many algorithms of Computer Algebra, particularly in the GröbnerCover and other algorithms for parametric polynomial systems. In this paper we review the canonical form ofconstructible sets and give algorithms for computing it.Peer ReviewedPostprint (author's final draft

A CDCL-style calculus for solving non-linear constraints

In this paper we propose a novel approach for checking satisfiability of non-linear constraints over the reals, called ksmt. The procedure is based on conflict resolution in CDCL style calculus, using a composition of symbolical and numerical methods. To deal with the non-linear components in case of conflicts we use numerically constructed restricted linearisations. This approach covers a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. A prototypical implementation has been evaluated on several non-linear SMT-LIB examples and the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at <http://informatik.uni-trier.de/~brausse/ksmt/

A New Algorithm for Computing Comprehensive Gröbner Systems

A new algorithm for computing a comprehensive Gröbner system of a parametric polynomial ideal over k[U][X] is presented. This algorithm generates fewer branches (segments) compared to Suzuki and Sato’s algorithm as well as Nabeshima’s algorithm, resulting in considerable efficiency. As a result, the algorithm is able to compute comprehensive Gröbner systems of parametric polynomial ideals arising from applications which have been beyond the reach of other well known algorithms. The starting point of the new algorithm is Weispfenning’s algorithm with a key insight by Suzuki and Sato who proposed computing first a Gröbner basis of an ideal over k[U, X] before performing any branches based on parametric constraints. Based on Kalkbrener’s results about stability and specialization of Gröbner basis of ideals

New computation method for stability conditions of Gröbner bases (Computer Algebra : Foundations and Applications)

Stability conditions of Gröbner bases are considered in the context of symbolic computation. New strategies for computing the stabitity conditions are introdcued. Moreover, a new algorithm for computing comprehensive Gröbner systems is given as the application