Kreuzer, Martin Robbiano, Lorenzo
Computational Linear and Commutative Algebra
Springer, Cham, 2016. xviii+321 pp. ISBN 978-3-319-43599-2; 978-3-319-43601-2
AMS Classification: 13P10 (13P15 15A27)

The first three chapters of this book give an advanced linear algebra course studying endomorphisms of finite-dimensional vector spaces over arbitrary fields. The authors call it linear algebra 1.5. It is explained that one does not need eigenvalues to define the theory of eigenspaces and generalized eigenspaces. The eigenvalues are replaced by so-called eigenfactors, the irreducible factors of the characteristic polynomial. An endomorphism with characteristic polynomial equal to the minimal polynomial gets a new name—they are called commendable and as usual play a special role. The theory of families of endomorphisms commuting pairwise is developed. From the point of view of commutative algebra, such a family is a finitely generated commutative algebra over the base field. For an ideal in this algebra, the kernel in the intersection of all kernels of endomorphisms is the ideal. A commuting family has joint eigenspaces (kernels of its maximal ideals) and joint generalized eigenspaces (kernels of the primary components of the zero ideal). The concept of commendability is transferred to families. It is proved that a family is commendable if and only if the vector space is a cyclic module with respect to the dual family.

The last three chapters of the book study special subjects of computational commutative algebra. A zero-dimensional affine algebra R over the base field is identified with a commuting family via its multiplication family F. Among other results, the authors prove that F is commendable if and only if R is Gorenstein. Several tools are developed for computing primary components of a zero-dimensional ideal.

The last chapter shows how to solve polynomial systems of equations over finite fields and over the field of rational numbers using joint eigenspaces, eigenvalues and eigenvectors.

The book is a textbook for advanced undergraduate and for graduate courses. Surprisingly, the experienced reader will also find new and unexpected aspects. I like the humorous style of the authors. The funny quotations help one enjoy the topic.

Reviewed by Gerhard Pfister