On coefficients of polynomials over finite fields

Abstract

In this paper we study the relation between coefficients of a polynomial over finite field F q and the moved elements by the mapping that induces the polynomial. The relation is established by a special system of linear equations. Using this relation we give the lower bound on the number of nonzero coefficients of polynomial that depends on the number m of moved elements. Moreover we show that there exist permutation polynomials of special form that achieve this bound when m | q − 1 . In the other direction, we show that if the number of moved elements is small then there is an recurrence relation among these coefficients. Using these recurrence relations, we improve the lower bound of nonzero coefficients when m ∤ q − 1 and m ⩽ q − 1 2 . As a byproduct, we show that the moved elements must satisfy certain polynomial equations if the mapping induces a polynomial such that there are only two nonzero coefficients out of 2 m consecutive coefficients. Finally we provide an algorithm to compute the coefficients of the polynomial induced by a given mapping with O ( q 3 / 2 ) operations.