100 Degree Function

Number Theory Level 5

Suppose $f(x)$ is a polynomial with integer coefficients of degree $100$ . Find the biggest possible number of pairs of integers $n<m$ , such that $f(n)=m$ and $f(m)=n$ .

Details and assumptions

You are asked to find the biggest possible number of pairs, not the biggest pair. Hence, your answer is just an integer, not a pair of integers.