The Gröbner cover
Gröbner bases turned out to be a powerful tool to study polynomial systems. The aim of the book is to study polynomial systems with parameters. \textit{V. Weispfenning} [J. Symb. Comput. 14, No. 1, 1--29 (1992; Zbl 0784.13013)] introduced comprehensive Gröbner bases, a tool which can handle polynomial systems with parameters. Let $K$ be a field and $\overline{K}$ its algebraic closure. Let $u=(u_1, \ldots, u_m)$ a set of parameters and $x=(x_1, \ldots, x_n)$ a set of variables. Let $I\subseteq K[u][x]$ be an ideal and for $a\in \overline{K}^m$ let $I_a$ be the image of $I$ in $\overline{K}[x]$ substituting $u=a$. A comprehensive Gröbner basis is a finite subset $G\subset I$ that specializes for every $a\in \overline{K}^m$ to a Gröbner basis of $I_a$. The Gröbner cover (introduced by [\textit{A. Montes} and \textit{M. Wibmer}, J. Symb. Comput. 45, No. 12, 1391--1425 (2010; Zbl 1207.13018)]) is the analog to reduced Gröbner bases of an ideal for parametric ideals. There is a canonical partition $\overline{K}^m=\overset{i}{\underset{i=1}{\bigcup}}S_i$ into disjoint locally closed subsets. For each $S_i$ there exists a generalized Gröbner basis, represented by a set of polynomial functions $B_i$ that specialize for $a\in S_i$ to the reduced Gröbner basis of $I_a$. If the ideal $I$ is homogeneous with respect to $x$ the $S_i$ are characterized by fixing the leading ideal of $I_a$. The Gröbner cover is the set $\{(S_1, B_1), \ldots, (S_r, B_r)\}$. An algorithm to compute the Gröbner cover was implemented by the author of the book 2010 as a library in the computer algebra system \textsc{Singular}. The book contains two parts, a theoretical part and a part with applications. The first chapter gives some basic results about Gröbner bases. Chapter 2, 3 and 4 contain basic tools needed for the construction of the Gröbner cover. In chapter 2 constructible sets are explained and algorithms to compute them. Chapter 3 introduces comprehensive Gröbner systems and bases and discusses different algorithms to compute them, starting with the algorithm of Suzuki-Sato to the actual algorithm of Kapur-Sun-Wang. Chapter 4 introduces $I$-regular functions, functions locally defined by polynomials, needed for the Gröbner cover. In chapter 5 the Gröbner cover is introduced and algorithms to compute it are discussed. All chapters contain a lot of examples. The algorithms discussed in chapter 2, 3, 4, 5 are implemented in the \textsc{Singular} library \texttt{grobcov.lib} and explicite examples to use the library are given. This is also true for the chapters in part II of the book. Here the last two chapters contain even exercises. Chapter 6 of the book is devoted to automatic deduction of geometric theorems. One example here is the nine point circle of Euler and Feuerbach, a circle passing through the feet of the heights of a triangle. Euler proved that it passes also through the midpoints of the sides of a triangle. Feuerbach proved that is passes also through the midpoints of the segments that join the vertices and the orthocenter. It is explained how to use the Gröbner cover to obtain this theorem automatically. Several other problems are treated in this chapter. Chapter 7 deals with the computation of geometric loci. One of the examples is Pascal's Limacon: Let $0$ be a fixed point on a circle $c$. Let $l$ be a line passing through $0$ and the mover point $M$ that glides on $c$. Let $T$ be the tracer point located on $l$ at a fixed distance from $M$. Pascal's Limacon is the locus of points $T$ as $M$ moves on the circle $c$. The problem leads to a system of three quadratic equations. It is shown how the Gröbner cover can be used to analyze this example. Chapter 8 studies geometric envelopes. In terms of loci and envelope computation the Gröbner cover can be used to introduce a taxonomy of the components and compute it. The definition of the envelope of a family of hypersurfaces is generalized and algorithms are given for its computation. The book contains an appendix explaning the buildtree algorithm used to obtain a disjoint reduced comprehensive Gröbner system.
scientific article; zbMATH DE number 6999636