Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result

Abstract

In this paper, we prove that for any given n ⩾ 2 , there exists a constant C ( n ) such that for any prime power q > C ( n ) , there exists a primitive normal polynomial of degree n over F q with the first ⌊ n 2 ⌋ coefficients prescribed, where the first coefficient is nonzero. This result strengthens the asymptotic result of the existence of primitive polynomials with the first ⌊ n − 1 2 ⌋ coefficients prescribed [S.Q. Fan, W.B. Han, p-Adic formal series and Cohen's problem, Glasg. Math. J. 46 (2004) 47–61] in two aspects. One is that we discuss in this paper not only the primitivity but also the normality. Another is that the number of the prescribed coefficients increases from ⌊ n − 1 2 ⌋ to ⌊ n 2 ⌋ . The estimates of character sums over Galois rings, the p-adic method introduced by the first two authors, and the computation technique used in [S.Q. Fan, W.B. Han, Primitive polynomial with three coefficients prescribed, Finite Fields Appl. 10 (2004) 506–521; D. Mills, Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl. 4 (2004) 1–22] are the main tools to get the above result.