Arithmetic constraints of polynomial maps through discrete logarithms

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