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1.
In this paper, we prove that for any given n2, 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 first coefficients prescribed, where the first coefficient is nonzero. This result strengthens the asymptotic result of the existence of primitive polynomials with the first 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 to . 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.  相似文献
2.
Primitive polynomial with three coefficients prescribed   总被引:1,自引:1,他引:0  
The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a1,a2,a3)Fq3, there exists a primitive polynomial f(x)=xn−σ1xn−1++(−1)nσn over Fq of degree n with the first three coefficients σ123 prescribed as a1,a2,a3 when n8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over Fq with the first three coefficient prescribed where the characteristic of Fq is 2 or 3.  相似文献
3.
Primitive normal polynomials with the last half coefficients prescribed   总被引:1,自引:0,他引:1  
In this paper, we prove that for any given n2, 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].  相似文献
4.
Primitive normal polynomials with a prescribed coefficient   总被引:1,自引:0,他引:1  
In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aFq and any 1m<n, there exists a primitive normal polynomial f(x)=xnσ1xn−1++(−1)n−1σn−1x+(−1)nσn such that σm=a, with the only exceptions σ1≠0. The theory can be extended to polynomials of smaller degree too.  相似文献
5.
《Acta Mathematica Hungarica》2008,118(1-2):129-148
In earlier papers, for “large” (but otherwise unspecified) subsets A, B of Z p and for h(x) ∈ Z p [x], Gyarmati studied the solvability of the equations a + b = h(x), resp. ab = h(x) with aA, bB, xZ p , and for large subsets A, B, C, D of Z p Sárk?zy showed the solvability of the equations a + b = cd, resp. ab + 1 = cd with aA, bB, cC, dD. In this series of papers equations of this type will be studied in finite fields. In particular, in Part I of the series we will prove the necessary character sum estimates of independent interest some of which generalize earlier results. Research partially supported by Hungarian National Foundation for Scientific Research, Grants No. T043623, T043631 and T049693.  相似文献
6.
We obtain an asymptotic formula for the -th power mean of odd primitive character sums over the interval .

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7.
In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

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8.
In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.

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9.
An improvement of Weil bound for a class of polynomials over GF(2n) is obtained.  相似文献
10.
We prove results on the distribution of points in an orbit of PGL(2,q) acting on an element of GF(qn). These results support a conjecture of Klapper. More precisely, we show that the points in an orbit are uniformly distributed if n is small with respect to q.  相似文献
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