A fundamental fact about Diophantine equations is that there can be no algorithm determining whether a given equation is soluble in integers Z or not. This is the famous negative solution of Hilbert's tenth problem by M. Davies, H. Putnam, J. Robinson, Ju. Matijasevič and G. Čudnovskii. More precisely, there exists a polynomial f(t; x1, …, xn) with integer coefficients such that there is no algorithm that would tell us whether for an integer t the equation f(t; x1, …, xn) = 0 is soluble in integers or not. The polynomial f(t; x, 1, …, xn) can be made explicit, for instance, we can have n = 13 (see, for example, [Manin, L], VI).

In this book, however, we are mostly interested in the solubility of Diophantine equations in the field of rational numbers Q and more general number fields. In this case the analogue of Hilbert's tenth problem is still open. For homogeneous equations the existence of solutions in Z and in Q is, of course, equivalent provided one does not count the all-zero solution.

For certain classes of equations an algorithm deciding the solubility over Q can be found. Such is the case when a class of projective varieties defined over Q satisfies the Hasse principle. This principle consists in requiring that the obvious necessary conditions for the solubility of a system of homogeneous polynomial equations with integer coefficients Fi(x1, … xn) = 0, i = 1, …, m, that is, the solubility of congruences modulo all the powers of prime numbers, and the solubility in the field of real numbers R, be also sufficient.