Estimation of multiple exponential sums associated with quartic polynomials

Let p be a prime number and f (x, y) be a polynomial in Zp[x, y]. For α > 1, the exponential sums associated with f modulo a prime pα is defined as S( f ; pα ) =Ι:x, y mod pα e α ( f (x, y)) . Estimation of S( f ; pα ) has been shown to depend on the cardinality of common roots of the partial derivative polynomials, fₓ and fy of f . Such cardinality then has been shown can be derived from the p-adic orders of common roots of the partial derivative polynomials, fₓ and fy in the neighbourhood of (x₀, y₀). The objective of this research is to arrive at such estimations associated with three different quartic polynomials. To achieve this objective we employ the Newton polyhedron technique to estimate the p-adic sizes of common zeros of partial derivative polynomials associated with the three quartic polynomials considered. The combination of indicator diagrams associated with the polynomials are examined and analyzed on cases where p-adic sizes of common zeros occur at the intersection point of the indicator diagrams. In addition, we apply certain conditions to ensure the existence of common zeros of partial derivative polynomials associated with the three quartic polynomials considered. The information obtained above is then applied to estimate the cardinality of the set α V ( f x , f y ; p ) . This estimation is then applied in turn to arrive at the estimation of exponential sums for the polynomials considered