Some authors restrict p(n) to be a linear function, but a more common definition is to allow p(n) to be any polynomial.
The complexity class EXPSPACE-complete is also a set of decision problems. A decision problem is in EXPSPACE-complete if it is in EXPSPACE, and every problem in EXPSPACE has a many-one reduction to it. In other words, there is a polynomial-time algorithm that transforms instances of one to instances of the other with the same answer. EXPSPACE-complete might be thought of as the hardest problems in EXPSPACE.
An example of an EXPSPACE-complete problem is the problem of recognizing whether two regular expressions represent different languages, where the expressions are limited to four operators: union, concatenation, the Kleene star (zero or more copies of an expression), and squaring (two copies of an expression).
If the Kleene star is left out, then that problem becomes NEXPTIME-complete, which is like EXPTIME-complete, except it is defined in terms of non-deterministic Turing machines rather than deterministic.