On the Estimate to Solutions of Congruence Equations Associated with a Cubic Form
The set of solutions to congruence equations modulo a prime power associated with the polynomial f(x,y) = ax3 + bx2 y + cxy2 + dt + kx + my + n in Z [x,y] is examined and its cardinality estimated by employing the Newton polyhedral technique. The method involves reduction of the partial derivatives...
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| Format: | Article |
| Language: | English English |
| Published: |
Universiti Putra Malaysia Press
1993
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| Online Access: | http://psasir.upm.edu.my/id/eprint/3085/1/On_the_Estimate_to_Solutions_of_Congruence.pdf |
| Summary: | The set of solutions to congruence equations modulo a prime power associated with the polynomial f(x,y) = ax3 + bx2 y + cxy2 + dt + kx + my + n in Z [x,y] is examined and its cardinality estimated by employing the Newton
polyhedral technique. The method involves reduction of the partial derivatives of f to single-variable polynomials and finding 8 the determinant factor in the estimation. f x and f yare reduced to one-variable polynomials by the
employment of suitable parameters. The Newton polyhedrons associated with the polynomials so obtained are then considered. There exist common zeros
of the single-variable polynomials whose p-adic orders correspond to the intersection points in the combination of the indicator diagrams associated
with the respective Newton polyhedrons. This leads to sizes of common zeros of the partial derivatives of f. This information is then used to arrive at the above estimate. |
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