Presentation of the Book The Gröbner Cover

The aim of this paper is to present the research book “The Gröbner Cover” Montes (The Gröbner Cover, ACM-series, Springer, Berlin, 2019) recently published. This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbner Cover, to the author’s best knowledge, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the automatic deduction of geometric theorems, loci computation and envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Gröbner Systems and Bases. It begins with Weispfenning introduction of Comprehensive Gröbner Systems (CGS), a fundamental contribution made in 1992, and provides a complete description of the Canonical Gröbner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed in Montes, Wibmer (J Symb Comput 45:1391–1425, 2010). In turn, the application part selects three problems for which the Gröbner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hyper-surfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described in the book have also been included in the Singular software library grobcov.lib implemented by the author and H. Schönemann, the book serving also as User Manual for the library.