Sains Malaysiana 36(1): 77-82 (2007)

 

 

Penganggaran Saiz p-Adic Pensifar Sepunya Terbitan

Separa Polinomial Berdarjah Enam

(Estimating the p-Adic Sizes of Common Zeros of Partial

Derivative  Polynomials of Degree Six)

 

 

S.H. Sapar &  K.A. Mohd Atan

Jabatan Matematik, Universiti Putra Malaysia

Laboratori Matematik Teori, Institut Penyelidikan Matematik

Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

 

 

ABSTRAK

 

Telah diketahui nilai hasil tambah eksponen S(f;pa) adalah bersandar kepada penganggaran bilangan unsur |V|, yang terdapat dalam set

dengan  menandakan polinomial-polinomial terbitan separa f terhadap . Kekardinalan bagi V pula bersandar kepada saiz  p-adic pensifar sepunya terbitan separa .

 

Makalah ini membentangkan suatu kaedah penentuan anggaran saiz p-adic bagi komponen (x,h) pensifar sepunya terbitan separa f(x,y) dalam  berdarjah enam berasaskan teknik polihedron Newton yang disekutukan dengan polinomial terbabit. Polinomial berdarjah enam yang dipertimbangkan berbentuk

 

.

 

Anggaran yang diperolehi adalah dalam sebutan saiz p-adic pekali-pekali sebutan yang dominan dalam f.

 

Kata kunci:  kekardinalan; pensifar sepunya; saiz p-adic; polihedron Newton; rajah penunjuk. 

 

ABSTRACT 

It is known that the value of the exponential sum S(f;pa) depends on the estimate of the cardinality |V|, the number of elements contained in the set

where  is the partial derivatives of f with respect to . The cardinality of V in turn depends on the p-adic sizes of common zeros of the partial derivatives .

 

This paper presents a method of determining the p-adic sizes of the components of (x,h) a common root of partial derivative polynomials of  f(x,y) in of degree six based on the p-adic Newton polyhedron technique associated with the polynomial. The degree six polynomial is of the form

 

 

The estimate obtained is in terms of the p-adic sizes of the coefficients of the dominant terms in  f.

 

Keywords :  cardinality; common zero;  p-adic sizes; Newton polyhedron; indicator diagram.

  

RUJUKAN/REFERENCS 

Chan K.L. dan. Mohd. Atan K.A 1997. On the Estimate to Solutions of Congruence Equations Associated with a Quartic Form.  Journal of  Physical Science 8:21-34.

Koblitz, N. 1977. p-adic Numbers, p-adic analysis and zeta Function. New York: Springer-Verlag.

Loxton J.H. dan Vaughn R.C. 1985. The Estimate of Complete Exponential Sums. Canad. Mth Bull. 28(4):440-454 .

Mohd. Atan K.A 1986. Newton Polyhedral Method of Determining p-adic Orders of Zeros Common to Two Polynomials in . Pertanika 9(3):375-380. Universiti Pertanian Malaysia.

Mohd. Atan K.A. dan Loxton J.H. 1986. Newton Polyhedra and Solutions of Congruences. In Loxton, J.H. and Van der Poorten, A.(ed). Diophantine Analysis. Cambridge : Cambridge University Press.

Mohd Atan K.A.dan Abdullah I.B 1992. Set of Solution to Congruences Equations Associated with Cubic Form. Journal of Physical Science 3:1-6.

Sapar S.H.dan Mohd Atan K.A. 2002. Penganggaran Kekardinalan Set Penyelesaian Persamaan Kongruen, Jurnal Teknologi C. 36:13-40.

 

 
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