Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique - Yap, Hong Keat (2010)
Let p be a prime and f (x, y) be a polynomial in Z [x, y] p . For α >1 , the exponential sums associated with f modulo a prime α p is defined as = Σ α α α y p pS f p e f x y , mod ( ; ) ( ( , )) . Estimation of ( ; ) α S f p has been shown to depend on the number and p-adic sizes of common roots of the partial derivative polynomials of f . The objective of this research is to arrive at such estimations associated with a quadratic and cubic polynomials f (x, y) . To achieve this objective we employ the p-adic methods and Newton polyhedron technique to estimate the p-adic sizes of common zeros of partial derivative polynomials associated with quadratic and cubic forms. The combination of indicator diagrams associated with the polynomials are examined and analyzed especially on cases where p-adic sizes of common zeros occur at the overlapping segments of the indicator diagrams. Cases involving p-adic sizes of common zeros that occur at simple points of intersection and the vertices have been investigated by earlier researchers. The information obtained above is then applied to estimate the cardinality of the set ( , ; ) α V f f p x y . This estimation is then applied in turn to arrive at the estimation of exponential sums for quadratic and cubic polynomials.