A Bound for the Number of Preimages of a Polynomial Mapping

An upper bound for the number of field elements that can be taken to roots of unity of fixed multiplicity by means of several given polynomials is obtained. This bound generalizes the bound obtained by V'yugin and Shkredov in 2012 to the case of polynomials of degree higher than $1$. This bound was obtained both over the residue field modulo a prime and over the complex field.