### ON THE CARDINALITY OF THE SET OF SOLUTIONS TO CONGRUENCE EQUATION ASSOCIATED WITH QUINTIC FORM

#### Abstract

The exponential sum associated with f is defined as

where the sum is taken over a complete set of residues modulo q and let x = (x1, x2, ... , xn) be a vector in the space Zn with Z ring of integers and q be a positive integer, f a polynomial in x with coefficients in Z. The value of S(f; q)

has been shown to depend on the estimate of the cardinality |jV|, the number of elements contained in the set

where fx is the partial derivative of f with respect to x = (x1, x2, ..., xn). This paper will give an explicit estimate of |V| for polynomial f(x; y) in Zp[x; y] of degree five. Earlier authors have investigated similar polynomials of lower

degrees. The polynomial that we consider in this paper is as follows:

The approach is by using p-adic Newton Polyhedron technique associated with this polynomial.

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