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 survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

Using Kapur-Sun-Wang algorithm for the Gröbner Cover

Kapur-Sun-Wang have recently developed a very efficient algorithm for computing Comprehensive Gröbner Systems that has moreover the required essential properties for being used as first step of the Gröbner Cover algorithm. We have implemented and adapted it inside the Singular grobcov library for computing the Gröbner Cover and there are evidences that it makes the canonical algorithm much more effective. In this note we discuss the performance of GC with KSW on a collection of examples.Peer ReviewedPostprint (published version

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure