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An Estimation of Exponential Sums Associated with a Cubic Form Polynomial

The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers. The main problem of the theory of exponential sums is to obtain an upper estimate of the modulus of an exponential sum as sharp as possible. Inve...

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Main Author: Heng, Swee Huay
Format: Thesis
Language:English
English
Published: 1999
Online Access:http://psasir.upm.edu.my/id/eprint/9506/1/FSAS_1999_45_A.pdf
http://psasir.upm.edu.my/id/eprint/9506/
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spelling my.upm.eprints.95062013-09-26T01:02:51Z http://psasir.upm.edu.my/id/eprint/9506/ An Estimation of Exponential Sums Associated with a Cubic Form Polynomial Heng, Swee Huay The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers. The main problem of the theory of exponential sums is to obtain an upper estimate of the modulus of an exponential sum as sharp as possible. Investigation on the sums when f is a two-variable polynomial is studied using the Newton polyhedron technique. One of the methods to obtain the estimate for the above exponential sums is to consider the cardinality of the set of solutions to congruence equations modulo a prime power. A closer look on the actual cardinality on the following polynomial in a cubic form f(x,y) = ax3 + bxi + cx + dy + e has been carried out using the Direct Method with the aid of Mathematica. We reveal that the exact cardinality is much smaller in comparison with the estimation. The necessity to find a more precise estimate arises due to this big gap. By a theorem of Bezout, the number of common zeros of a pair of polynomials does not exceed the product of the degrees of both polynomials. In this research, we attempt to find a better estimate for cardinality by looking at the maximum number of common zeros associated with the partial derivatives fx(x,y) and fy(x,y). Eventually a sharper estimate of cardinality for the various conditions on the coefficients of f(x,y) can be determined and the estimate of S(f; p') obtained. 1999-04 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/9506/1/FSAS_1999_45_A.pdf Heng, Swee Huay (1999) An Estimation of Exponential Sums Associated with a Cubic Form Polynomial. Masters thesis, Universiti Putra Malaysia. English
institution Universiti Putra Malaysia
building UPM Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Putra Malaysia
content_source UPM Institutional Repository
url_provider http://psasir.upm.edu.my/
language English
English
description The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers. The main problem of the theory of exponential sums is to obtain an upper estimate of the modulus of an exponential sum as sharp as possible. Investigation on the sums when f is a two-variable polynomial is studied using the Newton polyhedron technique. One of the methods to obtain the estimate for the above exponential sums is to consider the cardinality of the set of solutions to congruence equations modulo a prime power. A closer look on the actual cardinality on the following polynomial in a cubic form f(x,y) = ax3 + bxi + cx + dy + e has been carried out using the Direct Method with the aid of Mathematica. We reveal that the exact cardinality is much smaller in comparison with the estimation. The necessity to find a more precise estimate arises due to this big gap. By a theorem of Bezout, the number of common zeros of a pair of polynomials does not exceed the product of the degrees of both polynomials. In this research, we attempt to find a better estimate for cardinality by looking at the maximum number of common zeros associated with the partial derivatives fx(x,y) and fy(x,y). Eventually a sharper estimate of cardinality for the various conditions on the coefficients of f(x,y) can be determined and the estimate of S(f; p') obtained.
format Thesis
author Heng, Swee Huay
spellingShingle Heng, Swee Huay
An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
author_facet Heng, Swee Huay
author_sort Heng, Swee Huay
title An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
title_short An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
title_full An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
title_fullStr An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
title_full_unstemmed An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
title_sort estimation of exponential sums associated with a cubic form polynomial
publishDate 1999
url http://psasir.upm.edu.my/id/eprint/9506/1/FSAS_1999_45_A.pdf
http://psasir.upm.edu.my/id/eprint/9506/
_version_ 1643824334549024768
score 13.244404

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