Written mathematically, m(n) = O(nk) where k is a constant.
Mathematicians sometimes use the notion of "polynomial time on the length of the input" as a definition of a "fast" computation, as opposed to "super-polynomial time", which is anything slower than that. Exponential time is one example of a super-polynomial time.
The class of decision problems that can be solved on a deterministic sequential machine in polynomial time is known as P. The class of decision problems that can be verified in polynomial time is known as NP. Equivalently, NP is the class of decision problems that can be solved in polynomial time on a Non-deterministic Turing machine (NP stands for Nondeterministic Polynomial time).