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Reviewer: Albert Alkins Mullin

At the beginning of this century, David Hilbert, a leading mathematician, posed 23 enormously difficult technical problems that were intended to set the mathematical tone for the 20th century. He succeeded. The tenth problem (or tenth class of problems, since some of Hilbert's problems contain several very hard and largely unconnected problems worthy of separate consideration) is the only obvious decision problem among the 23 classes of problems; of course, any mathematical problem can be reformulated (somewhat artificially) as a decision problem. It asks for a single algorithm to test an arbitrary polynomial with integer coefficients for solvability in integers. The main theme of this book is to show that no such algorithm exists. The result was first proven in 1970 by the author. The implications of a negative solution to Hilbert's tenth problem are major not only in theory but in application. As an aside to the tenth problem, note that Hilbert's 23 classes of problems do not include either Fermat's last theorem (FLT) or the f our-color problem (now a computer-proven theorem). Hilbert may have concurred with Gauss's feelings concerning the lesser role of FLT itself in mathematics. It should be noted that, in a constructive sense, an individual subproblem of Hilbert's tenth problem exists that is logically equivalent to FLT. This monograph is well written and includes a broad spectrum of topics: exponentiation is diophantine; diophantine coding; primality is diophantine; universal diophantine equations; the recursive unsolvability of Hilbert's tenth problem; bounded universal quantifiers; diophantine complexity; and four number-theoretic appendices. Each chapter has exercises for the reader and commentary on historical perspective. Hints are provided for many exercises. I highly recommend this monograph to students and teachers of computer science, logic, and mathematics.