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
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Online Access:http://psasir.upm.edu.my/id/eprint/9506/1/FSAS_1999_45_A.pdf
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spelling my-upm-ir.95062013-09-26T01:02:51Z An Estimation of Exponential Sums Associated with a Cubic Form Polynomial 1999-04 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. Exponential sums Polynomials 1999-04 Thesis http://psasir.upm.edu.my/id/eprint/9506/ http://psasir.upm.edu.my/id/eprint/9506/1/FSAS_1999_45_A.pdf application/pdf en public masters Universiti Putra Malaysia Exponential sums Polynomials Faculty of Science and Environmental Studies English
institution Universiti Putra Malaysia
collection PSAS Institutional Repository
language English
English
topic Exponential sums
Polynomials

spellingShingle Exponential sums
Polynomials

Heng, Swee Huay
An Estimation of Exponential Sums Associated with a Cubic Form Polynomial
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
qualification_level Master's degree
author Heng, Swee Huay
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
granting_institution Universiti Putra Malaysia
granting_department Faculty of Science and Environmental Studies
publishDate 1999
url http://psasir.upm.edu.my/id/eprint/9506/1/FSAS_1999_45_A.pdf
_version_ 1747810968816058368