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Using the Teichmüller character and Gauss sums, we obtain the following results concerning p-ary bent functions and q-ary resilient functions: (1) a characterization of certain q-ary resilient functions in terms of their coefficients; (2) stronger upper bounds for the degree of p-ary bent functions; (3) determination of all bent functions on ; (4) a characterization of ternary weakly regular bent functions in terms of their coefficients.  相似文献
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Gauss和与广义Kloosterman和的几个新的恒等式   总被引：2，自引：0，他引：2

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Let N be the number of solutions of the equation
over the finite field . L. Carlitz found formulas for N when n=3 or 4, m1==mn=2. In an earlier paper, we obtained formulas for N when , where m=lcm[m1,…,mn], under a certain restriction on the exponents. In this paper, we find formula for N when md>2 and there exists an such that md|(p+1).  相似文献
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Let be the number of solutions of the equation over the finite field , and let be the number of solutions of the equation . If , let be the least integer represented by . and play important roles in estimating . Based on a partition of , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.
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Let be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials and . We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields of prime conductors and class numbers greater than or equal to . However, in accordance with Vandiver's conjecture, we found no example of for which divides .
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Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.  相似文献
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The classical incomplete Gauss sum
is studied for large N. An expansion is derived for this sum when p is an integer that holds uniformly for 1m<M0, M0=(N/p)1/(p-1) corresponding to the dominant, primary spiral when the terms are considered as unit vectors in the complex plane. This result is specialised to the quadratic incomplete Gauss sum (p=2) for which the spirals consist of regular traces dependent on the residue of N (mod 4). This expansion complements earlier work by Lehmer (Mathematika 23 (1976) 125) and extends the more recent results of Evans et al. (J. Math. Anal. Appl. 281 (2003) 454). The above expansion in the primary spiral is also discussed in the case p>1. Numerical results are given to demonstrate the accuracy of the various approximations.  相似文献
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We consider Gauss sums of the form

with a nontrivial additive character of a finite field of elements of characteristic . The classical bound becomes trivial for . We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on which is nontrivial for the values of of order up to . We also show that for almost all primes one can obtain a bound which is nontrivial for the values of of order up to .
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