Abstract
Let q be a prime power, let Fq be the finite field with q elements and let θ be a generator of the cyclic group Fq⁎. For each a∈Fq⁎, let logθa be the unique integer i∈{1,…,q−1} such that a=θi. Given polynomials P1,…,Pk∈Fq[x] and divisors 1<d1,…,dk of q−1, we discuss the distribution of the functionsFi:y↦logθPi(y)(moddi), over the set Fq∖∪i=1k{y∈Fq|Pi(y)=0}. Our main result entails that, under a natural multiplicative condition on the pairs (di,Pi), the functions Fi are asymptotically independent. We also provide some applications that, in particular, relates to past work.
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