Primitive normal polynomials with the last half coefficients prescribed

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 Fq with the last coefficients prescribed, where the last coefficient is a primitive element. Furthermore, the number of prescribed coefficients increases from to when the coefficients are specified as (0,…,0,b) with b any primitive element. This result is a complement to the existence of a primitive normal polynomial with the first coefficients prescribed which was proved in [S.Q. Fan, W.B. Han, K.Q. Feng, Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl. 13 (2007) 1029–1044]. The outline of this paper is similar to the above reference with the following two different treatments. On one hand, we use 1/x instead of x in the problem reduction step and as a consequence use the hybrid kloostermann sums instead of hybrid weil sums over Galois rings. On the other hand, the estimates are slightly more complicated and the results in some special cases are better than those in [S.Q. Fan, W.B. Han, K.Q. Feng, Primitive normal polynomials with multiple coefficients prescribed: An asymptotic result, Finite Fields Appl. 13 (2007) 1029–1044].