The terminology is certainly confusing. I think what you mean by “zero function” is the constant function on whose image is 0, not the zero polynomial in . In other words, and determine the same function on F iff considered as functions .
An example to show what I mean. Suppose , , . Then, considered as functions , and are equal (to the identity function on ), but as polynomials in .
In the example, the number of elements of is not greater than the degree of . What you are asked to show is that if , then as polynomials in if as functions .