The Distribution of Reciprocal Pairs Modulo Polynomials over a Finite Field

Abstract

AbstractGiven a finite field Fq of order q, a fixed polynomial g in –Fq[X] of positive degree, and two elements u and v in the ring of polynomials in R = Fq [X]/gFq[X], the question arises: How many pairs (a, 6) are there in R × R so that ab  1 mod g and so that a is close to u while b is close to v ? The answer is, about as many as one would expect. That is, there are no favored regions in R × R where inverse pairs cluster. The error term is quite sharp in most cases, being comparable to what would happen with random distribution of pairs. The proof uses Kloosterman sums and counting arguments. The exceptional cases involve fields of characteristic 2 and composite values of g. Even then the error term obtained is nontrivial. There is no computational evidence that inverses are in fact less evenly distributed in this case, however.