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A Formula for the Number of Solutions of Certain System of Special Equation over Finite Fields

Author: LuoYanMei
Tutor: CaoXiWang
School: Nanjing University of Aeronautics and Astronautics
Course: Applied Mathematics
Keywords: Finite fields diagonal equation Gauss sum Jacobi sum multiplicative character quadratic character
CLC: O153.4
Type: Master's thesis
Year: 2009
Downloads: 20
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Abstract


Polynomials over finite fields are among the most important parts in number theory, which play an important role in cryptography, coding theory and many other fields. From this observation, People start to devote more and more time into the research on polynomials, hence leads to the study of exponent equations over finite fields. In special cases one can give explicit formulas for the number of solution by restricting some conditions, but in general one will have to be satisfied with estimates. The paper is dedicated to the study of the number of solution of exponent equations a1x12+…+anxn2 =bx1…xs over finite fields. The main results are as follows:We shall first give a brief introduction about the research history of the number of solutions of exponent equations over finite fields and present some main results which have been obtained. In this paper we laconically work out an explicit formula for calculating the number of solutions of exponent equations a1x12+…+anxn2 =bx1…xs over finite fields where s is bigger than n with the use of theorems in exponential sums theory over finite fields, combinatorics and number theory and so on. We also adopt the methods from some papers in the references.Finally, we discuss three particular situations under condition of s < nand give three explicit formulas for calculating the number of solutions respectively by quadratic character and theory of exponential sums. These results not only enrich the research efforts on this equation by Ioulia,Baoulina, L.Carlitz etc. but also make a big progress in the study of the number of solution of exponent equations over finite fields.

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Abstract algebra ( Algebra ) > Field theory
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