Consider this example: a problem of dimension $n$ and $m$ ($m,n$: any given integers).has a search space of size $O(n^n * m^n)$. It is clear that this problem is exponential in $n$,whatsoever $m$ may be.My question: is this same problem polynomial in $m$? what are the assumptions if we can say that?is this way of complexity analysis correct? 
How to analyse the complexity of a problem with two or more size measures
complexity theory;time complexity;asymptotics