On exponential sums of x + λx with p ≡ e(mod d)

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

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On Waring’s problem: Three cubes and a sixth power

Nagoya Mathematical Journal ◽

10.1017/s0027763000007893 ◽

2001 ◽

Vol 163 ◽

pp. 13-53 ◽

Cited By ~ 4

Author(s):

Jörg Brüdern ◽

Trevor D. Wooley

Keyword(s):

Exponential Sums ◽

Large Integer ◽

Natural Numbers ◽

Waring’S Problem ◽

Smooth Numbers ◽

Waring's Problem ◽

Order Of Magnitude ◽

Almost All ◽

Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

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Construction of systems of polynomial equations with exact p-Divisibility via the covering method

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500133 ◽

2014 ◽

Vol 13 (06) ◽

pp. 1450013 ◽

Cited By ~ 3

Author(s):

Francis N. Castro ◽

Ivelisse M. Rubio

Keyword(s):

Exponential Sums ◽

Polynomial Equations ◽

Prime Field ◽

Elementary Method ◽

Covering Method ◽

Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

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