Estimation of Exponential Sums Using p-Adic 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 p-adic sizes of common roots...
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| Format: | Thesis |
| Language: | English English |
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2010
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| Online Access: | http://psasir.upm.edu.my/id/eprint/19679/ http://psasir.upm.edu.my/id/eprint/19679/1/IPM_2010_11_F.pdf |
| _version_ | 1848843793075273728 |
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| author | Yap, Hong Keat |
| author_facet | Yap, Hong Keat |
| author_sort | Yap, Hong Keat |
| building | UPM Institutional Repository |
| collection | Online Access |
| 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 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. |
| first_indexed | 2025-11-15T08:20:40Z |
| format | Thesis |
| id | upm-19679 |
| institution | Universiti Putra Malaysia |
| institution_category | Local University |
| language | English English |
| last_indexed | 2025-11-15T08:20:40Z |
| publishDate | 2010 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | upm-196792013-05-21T04:57:37Z http://psasir.upm.edu.my/id/eprint/19679/ Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique 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 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. 2010-12 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/19679/1/IPM_2010_11_F.pdf Yap, Hong Keat (2010) Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique. Masters thesis, Universiti Putra Malaysia. Newton diagrams p-adic analysis Estimation theory English |
| spellingShingle | Newton diagrams p-adic analysis Estimation theory Yap, Hong Keat Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique |
| title | Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique |
| title_full | Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique |
| title_fullStr | Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique |
| title_full_unstemmed | Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique |
| title_short | Estimation of Exponential Sums Using p-Adic Methods and Newton Polyhedron Technique |
| title_sort | estimation of exponential sums using p-adic methods and newton polyhedron technique |
| topic | Newton diagrams p-adic analysis Estimation theory |
| url | http://psasir.upm.edu.my/id/eprint/19679/ http://psasir.upm.edu.my/id/eprint/19679/1/IPM_2010_11_F.pdf |