# Non-effectivizable Estimates

Suppose that we have an equation

which for every value of the parameter *a* has at most finitely many solutions in *x,...,x**_{n}*. This fact can be expressed in two form:

- • Equation (2.33) has at most
*v(a)*solutions; - • in every solution of (2.33) xi <
*a (a)***,***x*_{n}*< a (a)*

for suitable functions *v* and *a*.

From a mathematical point of view these two statements are equivalent. However, they are rather different computationally. Having *a(a)* we can calculate *v(a)* but not *vice versa** .* Number-theorists have found many classes of Diophantine equations with computable

*v(a)*for which they fail to compute o(a). In such cases number-theorists say that “the estimate of the size of solutions is

*non-effective”.*

Now let us take some undecidable set *M* and construct an exponential Diophantine equation

giving a single-fold representation for *M*. Clearly, Eq. (2.34) has the following two properties:

- • for every value of the parameter a,Eq. (2.34) has at most one solution in
*x*_{1}, ...,x_{n}; - • for every effectively computable function
*о*there is a value of*a*for which the Eq. (2.34) has a solution*x*such that max{x_{1},...,x_{n}_{1},*...,x*_{n}}*> о (a)*(otherwise we would be able to determine whether*a*belongs to*M*or not).

In other words, the boundedness of solutions of equation (2.34) cannot be made effective in principle. This relationship between undecidability and non- effectivizability is one of the main stimuli to improve the DPRM-theorem to singlefold (or at least to finite-fold) representations and thus establish the existence of non-effectivizable estimates for genuine Diophantine equations.