A Note on Hilbert's Tenth Problem

Extract

The tenth problem on Hilbert's well known list [1] is the following.

(H 10) For an arbitrary polynomial P = P(x₁,x₂,…,x_n) with integer coefficients to determine whether or not the equation P = 0 has a solution in integers.

By 'integers' we always mean 'rational integers'. The problem (H 10) is still unsolved but it appears likely that no decision procedure exists; in this connection see [2]. It will be shown here that (H 10) is equivalent to deciding whether or not every member of a certain given countable sec of rational functions of a single variable x is absolutely monotonie. We recall that f(x) is absolutely monotonie in I if f(x) possesses non-negative derivatives of all orders at every x ∊ I.