Diophantine Equations and Computation

Abstract

Unless otherwise stated, we’ll work with the natural numbers: $$N = \{0,1,2,3, \dots\}.$$ Consider a Diophantine equation F(a1,a2,...,a n ,x1,x2,...,x m ) = 0 with parameters a1,a2,...,a n and unknowns x1,x2,...,x m For such a given equation, it is usual to ask: For which values of the parameters does the equation have a solution in the unknowns? In other words, find the set: $$ \{<a_1,\ldots,a_n> \mid \exists x_1,\ldots,x_m [F(a_1,\ldots,x_1,\ldots)=0] \}$$ Inverting this, we think of the equation F = 0 furnishing a definition of this set, and we distinguish three classes: a set is called Diophantine if it has such a definition in which F is a polynomial with integer coefficients. We write \(\cal D\) for the class of Diophantine sets. a set is called exponential Diophantine if it has such a definition in which F is an exponential polynomial with integer coefficients. We write \(\cal E\) for the class of exponential Diophantine sets. a set is called recursively enumerable (or listable)if it has such a definition in which F is a computable function. We write \(\cal R\) for the class of recursively enumerable sets.