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.