Cardinality of sets associated to certain degree seven polynomials
Let f = f(x,y) be a function of two variables. Let q be an integer and let S(f;q) = ∑x mod qe 2πif(x)/q, where the sum is taken over a complete set of residue modulo q. The value of S(f;q) depends on the estimate of the cardinality |V| of the following set V={(x,y)mod q |fx, fy ≡ 0 mod q} where fx a...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Faculty of Science, University of Malaya
2015
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| Online Access: | http://psasir.upm.edu.my/id/eprint/51969/ http://psasir.upm.edu.my/id/eprint/51969/1/Cardinality%20of%20sets%20associated%20to%20certain%20degree%20seven%20polynomials.pdf |
| Summary: | Let f = f(x,y) be a function of two variables. Let q be an integer and let S(f;q) = ∑x mod qe 2πif(x)/q, where the sum is taken over a complete set of residue modulo q. The value of S(f;q) depends on the estimate of the cardinality |V| of the following set V={(x,y)mod q |fx, fy ≡ 0 mod q} where fx and fy are the partial derivative of f with respect to x and y. In this paper, we discuss the cardinality, |V| of the set of solutions for congruence equations of some special binary forms. Firstly we need to obtain the p-adic sizes of common zeros of the partial derivative polynomials by using Newton polyhedron technique. The polynomial that we consider is in the form of f(x, y) = ax7 + bx6y + cx5y2 + sx + ty + k. |
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