Undecidable Diophantine Equations

In 1900 Hubert asked for an algorithm to decide the solvability of all diophantine equations, P(x1, . . . , xv) = 0, where P is a polynomial with integer coefficients. In special cases of Hilbert's tenth problem, such algorithms are known. Siegel [7] gives an algorithm for all polynomials P(xx, . . . , xv) of degree < 2. From the work of A. Baker [1] we know that there is also a decision procedure for the case of homogeneous polynomials in two variables, P(x, y) = c. The first steps toward the eventual negative solution of the entire (unrestricted) form of Hilbert's tenth problem, were taken in 1961 by Julia Robinson, Martin Davis and Hilary Putnam [2]. They proved that every recursively enumerable set, W can be represented in exponential diophantine form