On The Estimate to Solutions of Congruence Equations Associated with a Quartic form

The set of solutions to congruence equations modulo a prime power associated with the polynomial in Zp[x,y] is examined and its cardinality is estimated by employing the Newton polyhedral technique.The method involves reduction of the partial derivatives of f that is f" and fy into polynomials with single variable and finding 8 the determinant factor in the estimation. fx and fy are reduced to one-variable polynomials by employment of suitable parameters. The Newton polyhedrons associated with the polynomials so obtained are then considered and combination of their Indicator diagrams examined. 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 of the polynomials. The p-adic sizes of these zeros are then determined, and this leads to sizes of common zeros of the partial derivatives of f. This information is then used to arrive at the estimate of the cardinality above.

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