# Complexity of checking $a > br^m + cr^n$, with $r$ rational

I'm wondering if the following problem is decidable in P-time (or even NP):

Given $$a, b, c \in \mathbb{Z}$$ and $$m, n, p, q \in \mathbb{N}$$ all in binary, decide if $$a > br^m + cr^n$$, where $$r = {p \over q}$$.

Remarks:

1. If $$c = 0$$, this is a very special case of the following problem, which is in P for any fixed $$n$$ (Etessami et al.):

Given $$\vec{a}, \vec{b}, \vec{c}, \vec{d} \in \mathbb{N}^n$$ all in binary, decide if $$a_1^{b_1}\cdots a_n^{b_n} > c_1^{d_1}\cdots c_n^{d_n}$$.

1. An upper bound for the above problem is given by the fact that numerical analysis is in the counting hierarchy (Allender et al.)

2. I'm particularly interested in values of $$r$$ very close to 1.