Análogos del décimo problema de Hilbert = Analogues of hilbert's tenth problem.

Abstract

The tenth problem in D. Hilbert's famous list asked the following : Devise an algorithm to decide whether a polynomial equation with integer coe cients has an integer solution. (These equations are called Diophantine equations.) In the year 1970, 70 years after Hilbert posed it, Y. Matiyasevich (based on work of M. Davis, J. Robinson and H. Putnam) proved that such an algorithm does not exist [10]. Knowing that the decision problem for integer solutions of Diophantine equations had a negative answer, the problem shifted to smaller classes of equations. For example, it follows from Matiyasevich's negative answer that there exists no algorithm to decide whether a system of second-degree Diophantine equations has integral solutions; while, on the other hand, a result of M. Presburger (1929) implies that an analogous algorithm for systems of linear Diophantine equations exists [15]. Consider all systems of second degree Diophantine equations in where every unknown appears squared in all of its ocurrences. These systems form a subset of all systems of second degree Diophantine equations, and it can be shown that every linear Diophantine equation can be written as such (just because any integer can be written as x2+y2ð€€€z2 for some integers x, y and z). Thus, the decision problem for integer solutions to this kind of systems of equations is \in between" the two already known results mentioned in the previous paragraph. This problem is known as the Problem of representation by diagonal quadratic forms, and is currently open.