Fermat quotients over function fields (Q1266402)

The authors study the generalization of the classical Fermat quotients to rational function fields over finite fields: if \(P\) is an irreducible polynomial and \(A\) is any polynomial over \({\mathbb F}_{q}\), the Fermat quotient of \(A\) with respect to \(P\) is the polynomial \(Q_{P}(A) = {A^{\deg(P)} - A \over P}\). Several complete results about such quotients are obtained, while little is known for traditional Fermat quotients over the integers. For instance, they give the exact power of \(P\) dividing \(Q_{P}(A)\), and the exact size of the set \(\{ Q_{P}(A) : \deg A < \deg P \}\) considered modulo \(P\). The study of rational field Fermat quotients is reduced to that of quotients \({A^{\deg(P)} - A \over x^{q^{n}}-x}\), where \(n\) is a positive integer.