The Gröbner basis provides a canonical representation of ordinary polynomial systems. This development allowed computers to solve polynomial equations across mathematics, engineering, and other sciences; however, the ability to tackle polynomial systems with parameters was still lacking. This is where the Gröbner cover comes in. The author, together with other collaborators, developed the Gröbner cover to allow for computational solutions to parametric polynomial systems.

The book assumes that the reader is closely familiar with the ordinary Gröbner basis theory of multivariate polynomials. In fact, the following sentence is repeated twice in the text: “It is assumed that the reader is familiar with ordinary Gröbner basis theory.” If, like me, you are not, this book may prove an intellectual challenge. This has been the hardest book I have had the opportunity to review in more than 15 years of reviewing for CR. Despite the 50-odd hours invested, I still struggled to follow many of the leaps of intuition required by the explanations.

After a brief explanation of the notation used in chapter 1, the book splits into two parts. The first part follows the evolutionary steps that took the author from the Gröbner basis to the Gröbner cover. The second part of the book aims to show the usefulness of the approach by applying the theory to a wide variety of problems, each chapter covering a particular problem domain area. Given that you are still reading this review, let me now take you through a more detailed overview of each of the eight chapters.

Chapter 1 provides a brief summary of the theory that underpins the latter extensions. As mentioned previously, the first chapter is an opportunity to introduce the notation and style used throughout the book. Chapter 2 explains constructible sets, the construction of locally closed sets and their canonical representation. This chapter also introduces the reader to the grobcov library developed by the author within the Singular programming language.

Chapter 3 introduces parametric polynomial systems of equations. It defines specialization to be a ring homomorphism and shows how to represent polynomials under specialization. It then introduces three algorithms that are required as building blocks for computing comprehensive Gröbner systems and basis. Finally, the chapter describes the Kapur-Sun-Wang algorithm, a faster alternative to the BuildTree algorithm used by the author in prior iterations of grobcov.lib.

Chapter 4 defines I-regular functions on a locally closed set. These are required for establishing the theory behind the Gröbner cover. The chapter discusses both regular and I-regular functions and how to compute a full representation from a generic one, and in the process introduces three more algorithms. It is in chapter 5 that the reader finally gets to the theory behind the Gröbner cover. First, it uses homogeneous ideals to prove the existence of the canonical Gröbner cover, after which the theory is extended to arbitrary ideals. The chapter ends with a demonstration of the grobcov command within the Singular library.

Having covered the theory, the book changes direction and looks at applications. Chapter 6 looks into how to deduce geometric theorems automatically. Starting with the classical Steiner-Lehmus theorem (briefly stated as the inner bisectors of angles A and B of a triangle, A, B, and C are of equal lengths if and only if the triangle is isosceles), the author shows how to use the Gröbner cover to study the proposition and convert it into a theorem. This is then expanded to find the supplementary conditions required for a geometric proposition to become a theorem. The chapter provides further examples using an orthic triangle and the nine points of Euler’s circle.

Chapter 7 delves into geometric loci and their construction. More simply, a locus is the set of points satisfying some conditions. It defines multiple classes of problems and provides examples for each. Once a taxonomy of a locus is defined, algorithms are proposed that are able to classify the components and compute anti-images. The chapter ends with a description of the locus algorithm, which is then applied to a number of geometric problems.

The final chapter of the book extends the concepts developed for loci problems to envelopes of families of curves or surfaces. It includes many examples and provides computed and color images of the envelopes. It considers both open and closed envelopes.

In summary, I consider the book a highly technical text that could have been made more accessible to a wider audience.