An Estimation of Exponential Sums Associated with a Cubic Form Polynomial

The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers. The main problem of the theory of exponential sums is to obtain an upper estimate of the modulus of an exponential sum as sharp as possible. Investigation on the sums when f is a two-variable polynomial is studied using the Newton polyhedron technique. One of the methods to obtain the estimate for the above exponential sums is to consider the cardinality of the set of solutions to congruence equations modulo a prime power. A closer look on the actual cardinality on the following polynomial in a cubic form f(x,y) = ax3 + bxi + cx + dy + e has been carried out using the Direct Method with the aid of Mathematica. We reveal that the exact cardinality is much smaller in comparison with the estimation. The necessity to find a more precise estimate arises due to this big gap. By a theorem of Bezout, the number of common zeros of a pair of polynomials does not exceed the product of the degrees of both polynomials. In this research, we attempt to find a better estimate for cardinality by looking at the maximum number of common zeros associated with the partial derivatives fx(x,y) and fy(x,y). Eventually a sharper estimate of cardinality for the various conditions on the coefficients of f(x,y) can be determined and the estimate of S(f; p') obtained.

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