A Simple Polynomial for a Simple Transposition
Greg Martin · American Mathematical Monthly · 2008
We are used to thinking of polynomials as very special functions, and with good reason when the domain and range are the real numbers or the rational numbers. However, the situation can be counterintuitive over other rings: over finite fields, for example, every function is a polynomial! It can thus be interesting to look at familiar functions over less familiar rings and find out what sort of polynomial represents them. Given any ring R, we say that the polynomial P(x) ∈ R[x] represents the function f(x) : R → R if P(a) = f(a) for every a ∈ R. In a note in this Monthly, Chen and Mullen [1] considered when various permutations of the elements of Zn can be represented by polynomials over Zn. One of the remarks they made is that for odd primes p, the polynomial f(x) = − [ ( (x − 1) p−2 + 1) p−2] p−2