Citation
Lasaraiya, Suriana
(2016)
Exponential sums for some nth degree polynomial.
Masters thesis, Universiti Putra Malaysia.
Abstract
Let f(x, y) be a polynomial in Zp[x, y] and p be a prime. For α > 1, the exponential
sums associated with f(x, y) modulo a prime p
α is defined as S(f ; p
α) =
epα (f(x, y)), where the sum is taken over a complete set of residues modulo p
α. It
has been shown that the exponential sums is depends on the cardinality of the set
of solutions to the congruence equation associated with the polynomial f(x, y). The
objective of this research is to find an estimation of the exponential sums for some
n
th degree polynomial at any point (x−x0, y−y0). There are two conditions being
considered, that is for ordpb
2 6= ordpac and ordpb
2 = ordpac.
The padic methods and Newton polyhedron technique is used to estimate the padic
sizes of common zeros of partial derivative polynomials associated with n
th degree
polynomial, where n ≥ 3. Then, construct the combination of indicator diagram associated
with some n
th degree polynomial. The indicator diagram is then examined
and analyzed.
The information of padic sizes of common zeros that obtained is applied to estimate
the cardinality of the set V(fx, fy; p
α). The results of the cardinality is then used to
estimate the estimation of exponential sums associated to n
th degree polynomial,
where n ≥ 3.
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