Content of the book

# "HILBERT's TENTH PROBLEM"

written by Yuri MATIYASEVICH

Preface

1 Principal Definitions
1.1 Diophantine equations as a decision problem
1.2 Systems of Diophantine equations
1.3 Solutions in natural numbers
1.4 Families of Diophantine equations
1.5 Logical terminology
1.6 Some simple examples of Diophantine sets, properties, relations, and functions
Exercises
Commentary

2 Exponentiation Is Diophantine
2.1 Special second-order recurrent sequences
2.2 The special recurrent sequences are Diophantine (basic ideas
2.3 The special recurrent sequences are Diophantine (proof)
2.4 Exponentiation is Diophantine
2.5 Exponential Diophantine equations
Exercises
Commentary

3 Diophantine Coding
3.1 Cantor numbering
3.2 Gödel coding
3.3 Positional coding
3.4 Binomial coefficients, the factorial, and the prime numbers are Diophantine
3.5 Comparison of tuples
3.6 Extensions of functions to tuples
Exercises
Commentary

4 Universal Diophantine Equations
4.1 Basic definitions
4.2 Coding equations
4.3 Coding possible solutions
4.4 Computing the values of polynomials
4.5 Universal Diophantine equations
4.6 Diophantine sets with non-Diophantine complements
Exercises
Commentary

5 Hilbert's Tenth Problem Is Unsolvable
5.1 Turing machines
5.2 Composition of machines
5.3 Basis machines
5.4 Turing machines can recognize Diophantine sets
5.5 Diophantine simulation of Turing machines
5.6 Hilbert's Tenth Problem is undecidable by Turing machines
5.7 Church's Thesis
Exercises
Commentary

6 Bounded Universal Quantifiers
6.1 First construction: Turing machines
6.2 Second construction: Gödel coding
6.3 Third construction: summation
6.4 Connections between Hilbert's Eighth and Tenth Problems
6.5 Yet another universal equation
6.6 Yet another Diophantine set with non-Diophantine complement
Exercises
Commentary

7 Decision Problems in Number Theory
7.1 The number of solutions of Diophantine equations
7.2 Non-effectivizable estimates in the theory of exponential Diophantine equations
7.3 Gaussian integer counterpart of Hilbert's Tenth Problem
7.4 Homogeneous equations and rational solutions
Exercises
Commentary

8 Diophantine Complexity
8.1 Principal definitions
8.2 A bound for the number of unknowns in exponential Diophantine representations
Exercises
Commentary

9 Decision Problems in Calculus
9.1 Diophantine real numbers
9.2 Equations, inequalities, and identities in real variables
9.3 Systems of ordinary differential equations
9.4 Integrability
Exercises
Commentary

10 Other Applications of Diophantine Representations
10.1 Diophantine games
10.2 Generalized knights on a multidimensional chessboard
Exercises
Commentary

Appendix
1 The Four Square Theorem
2 Chinese Remainder Theorem
3 Kummer's Theorem
4 Summation of a generalized geometric progression

Hints to the Exercises

Bibliography

List of Notation

Name Index

Subject Index