An explicit estimation of exponential sum associated with a quintic form

It has been shown that the estimation of an exponential sum associated with a two-variable polynomial can be derived from the estimation of the cardinality of the set of common solutions to congruence equations associated with the partial derivatives of the polynomial. By an application of this principle we will present in this talk a method of determining an exponential sum associated with a quintic of the form f(x,y)=ax5 + bx4y + cx3y2 + tx + sy + m over the ring Zp where p is an odd prime. An explicit estimate of the cardinality of the set of solutions common to the partial derivative polynomials fx and fy associated with f(x, y) will be first determined by examining the indicator diagrams associated with the Newton polyhedra of both polynomials. By applying results of our earlier works we will derive an explicit estimation of the upper bound of the sum in terms of the p-adic sizes of some invariants associated with the polynomial.

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