The theory of polynomial identities, as a well-defined field of study, began with a well-known 1948 article of Kaplansky. The field has since developed along two branches: the structural, which investigates the properties of rings which satisfy a polynomial identity; and the varietal, which investigates the set of polynomials in the free ring which vanish under all specializations in a given ring. This book is based on lectures delivered during an NSF-CBMS Regional Conference, held at DePaul University in July 1990, at which the author was the principal lecturer. The first part of the book is concerned with polynomial identity rings. The emphasis is on those parts of the theory related to n x n matrices, including the major structure theorems and the construction of certain polynomials identities and central polynomials for n x n matrices. The ring of generic matrices and its centre is described. The author then moves on to the invariants of n x n matrices, beginning with the first and second fundamental theorems, which are used to describe the polynomial identities satisfied by n x n matrices. One of the exceptional features of this book is the way it emphasizes the connection between polynomial identities and invariants of n x n matrices. Accessible to those with background at the level of a first-year graduate course in algebra, this book gives readers an understanding of polynomial identity rings and invariant theory, as well as an indication of current problems and research in these areas.

Computational Aspects of Polynomial Identities: Volume l, Kemer's Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer's proof of Specht's conjecture for affine PI-algebras in characteristic 0.The

The Handbook of Geometric Constraint Systems Principles is an entry point to the currently used principal mathematical and computational tools and techniques of the geometric constraint system (GCS). It functions as a single source containing the core principles and results, accessible to both beginners and experts. The handbook provides a guide for students learning basic concepts, as well as experts looking to pinpoint specific results or approaches in the broad landscape. As such, the editors created this handbook to serve as a useful tool for navigating the varied concepts, approaches and results found in GCS research. Key Features: A comprehensive reference handbook authored by top researchers Includes fundamentals and techniques from multiple perspectives that span several research communities Provides recent results and a graded program of open problems and conjectures Can be used for senior undergraduate or graduate topics course introduction to the area Detailed list of figures and tables About the Editors: Meera Sitharam is currently an Associate Professor at the University of Florida’s Department of Computer & Information Science and Engineering. She received her Ph.D. at the University of Wisconsin, Madison. Audrey St. John is an Associate Professor of Computer Science at Mount Holyoke College, who received her Ph. D. from UMass Amherst. Jessica Sidman is a Professor of Mathematics on the John S. Kennedy Foundation at Mount Holyoke College. She received her Ph.D. from the University of Michigan.

The book is devoted to the combinatorial theory of polynomial algebras, free associative and free Lie algebras, and algebras with polynomial identities. It also examines the structure of automorphism groups of free and relatively free algebras. It is based on graduate courses and short cycles of lectures presented by the author at several universities and its goal is to involve the reader as soon as possible in the research area, to make him or her able to read books and papers on the considered topics. It contains both classical and contemporary results and methods. A specific feature of the book is that it includes as its inseparable part more than 250 exercises and examples with detailed hints (50 % of the numbered statements), some of them treating serious mathematical results. The exposition is accessible for graduate and advanced undergraduate students with standard background on linear algebra and some elements of ring theory and group theory. The professional mathematician working in the field of algebra and other related topics also will find the book useful for his or her research and teaching. TOC:Introduction 1. Commutative, Associative and Lie Algebras: Basic properties of algebras; Free algebras; The Poincaré-Birkhoff-Witt theorem. 2. Algebras with Polynomial Identities: Definitions and examples of PI-Algebras; Varieties and relatively free algebras; The theorem of Birkhoff. 3. The Specht Problem: The finite basis property; Lie algebras in characteristic 2. 4. Numerical Invariants of T-Ideals: Graded vector spaces; Homogeneous and multilinear polynomial identities; Proper polynomial identities. 5. Polynomial Identities of Concrete Algebras: Polynomial identities of the Grassmann algebra; Polynomial identities of the upper triangular matrices. 6. Methods of Commutative Algebra: Rational Hilbert series; Nonmatrix polynomial identities; Commutative and noncommutative invariant theory. 7. Polynomial Identities of the Matrix Algebras: The Amitsur-Levitzki theorem; Generic matrices; Central polynomials; Various identities of matrices. 8. Multilinear Polynomial Identities: The codimension theorem of Regev; Algebras with polynomial growth of codimensions; The Nagata-Higman theorem; The theory of Kemer. 9. Finitely Generated PI-Algebras: The problems of Burnside and Kurosch; The Shirshov theorem; Growth of algebras and Gelfand-Kirillov dimension; Gelfand-Kirillov dimension of PI-Algebras. 10. Automorphisms of Free Algebras: Automorphisms of groups and algebras; The polynomial algebra in two variables; The free associative algebra of rank two; Exponential automorphisms; Automorphisms of relatively free algebras. 11. Free Lie Algebras and Their Automorphisms: Bases and subalgebras of free Lie algebras; Automorphisms of free Lie algebras; Automorphisms of relatively free Lie algebras. 12. The Method of Representation Theory: Representations of finite groups; The symmetric group; Multilinear polynomial identities; The action of the general linear group; Proper polynomial identities; Polynomial identities of matrices.

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

This book is intended for graduate students and researchers interested in the mathematical physics and PDE.