How well can we do with this variable assignment problem?
Assuming that the $c_{a,b}$ values appears in a nicely ordered way, you can use Horner's rule to evaluate the polynomial in $O(n^2)$ arithmetic operations. As mentioned by frogeyedpeas, you can't expect beating $n^2$ because it is the size of the input.

Note that if you are interested by the number of bit operations then you need to multiply this by the elementary cost of multiplication (for numbers of the size considered in your problem).

It you want to beat the $n^2$ bound, you need the $c_{a,b}$ to be represented in a more compact/structured way.