In the beginning

The subject of this book dates back to the beginning of the twentieth century. In 1900, at the International Congress of mathematicians, David Hilbert presented a list of problems, which exerted great influence on the development of mathematics in the twentieth century. The tenth problem on the list had to do with solving Diophantine equations. Hilbert was interested in the construction of an algorithm which could determine whether an arbitrary polynomial equation in several variables had solutions in the integers. If we translate Hilbert's question into modern terms, we can say that he wanted a program taking coefficients of a polynomial equation as input and producing a “yes” or “no” answer to the question “Are there integer solutions?” This problem became known as Hilbert's Tenth Problem (HTP).

It took some time to prove that the algorithm requested by Hilbert did not exist. At the end of the sixties, building on the work of Martin Davis, Hilary Putnam, and Julia Robinson, Yuri Matiyasevich proved that Diophantine sets over Z were the same as recursively enumerable sets and, thus, that Hilbert's Tenth Problem was unsolvable. The original proof and its immediate implications have been described in detail. The reader is referred to, for example, a book by Matiyasevich (see [52] – the original Russian edition – or [53], an English translation), an article by Davis (see [12]) or an article by Davis, Matiyasevich, and Robinson (see [14]).