Exponential sums for eighth degree polynomial - Low, Chee Wai; Sapar, Siti Hasana; Mohamat Johari, Mohamat Aidil (Institute for Mathematical Research, Universiti Putra Malaysia, 2020)
Let p > 7 be a prime, the exponential sums of any polynomial f(x, y) is given by S(f; p α ) = ∑x,y mod p e 2πif(x,y)/ pα, where the sum is taken over a complete set of residue modulo p. Firstly, Newton Polyhedron technique was used to determine the estimation for the p-adic sizes of common zeros of the partial derivative polynomials fx, fy which derive from f(x, y). We continue by estimating the cardinality N(g, h; p α ) as well as the exponential sums of polynomial f(x, y). Throught out this paper, we consider the polynomial of eighth degree with two variables in the form f(x, y) = ax8 +bx7 y+cx6 y 2 +dx5 y 3 +ex4 y 4 +kx3 y 5 +mx2 y 6 + nxy7 + uy8 + rx + sy + t.