Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique
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 padic sizes of common roots...
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myupmir.1967920130521T04:57:37Z Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 201012 Yap, Hong Keat 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 padic 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 padic methods and Newton polyhedron technique to estimate the padic 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 padic sizes of common zeros occur at the overlapping segments of the indicator diagrams. Cases involving padic 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. Newton diagrams padic analysis Estimation theory 201012 Thesis http://psasir.upm.edu.my/id/eprint/19679/ http://psasir.upm.edu.my/id/eprint/19679/1/IPM_2010_11_F.pdf application/pdf en public masters Universiti Putra Malaysia Newton diagrams padic analysis Estimation theory Institute for Mathematical Research English 
institution 
Universiti Putra Malaysia 
collection 
PSAS Institutional Repository 
language 
English English 
topic 
Newton diagrams padic analysis Estimation theory 
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Newton diagrams padic analysis Estimation theory Yap, Hong Keat Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 
description 
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 padic 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 padic methods and Newton polyhedron technique to estimate the padic 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 padic sizes of common zeros occur at the overlapping segments of the indicator diagrams. Cases involving padic 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. 
format 
Thesis 
qualification_level 
Master's degree 
author 
Yap, Hong Keat 
author_facet 
Yap, Hong Keat 
author_sort 
Yap, Hong Keat 
title 
Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 
title_short 
Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 
title_full 
Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 
title_fullStr 
Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 
title_full_unstemmed 
Estimation of Exponential Sums Using pAdic Methods and Newton Polyhedron Technique 
title_sort 
estimation of exponential sums using padic methods and newton polyhedron technique 
granting_institution 
Universiti Putra Malaysia 
granting_department 
Institute for Mathematical Research 
publishDate 
2010 
url 
http://psasir.upm.edu.my/id/eprint/19679/1/IPM_2010_11_F.pdf 
_version_ 
1747811440874487808 