Sains Malaysiana 40(8)(2011): 921–926

 

Estimation of p-adic sizes of Common Zeros of Partial Derivatives Associated with a Cubic Form

(Penganggaran Saiz p-Adic Pensifar Sepunya Polinomial-polinomial Terbitan Separa disekutukan dengan Bentuk Kubik)

 

H.K. Yap*, K.A. Mohd Atan & S.H. Sapar

Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang

Selangor, Malaysia

 

Diserahkan: 13 November 2009 / Diterima: 17 Januari 2011

 

ABSTRACT

In this paper we determined the estimate of p-adic sizes of common zeros of partial derivative polynomials associated with a cubic form whose indicator diagrams have one overlapping segment by using Newton polyhedron technique. We showed that the p-adic sizes of such common zeros can be found explicitly on the overlapping segment of the indicator diagrams associated with the polynomials.

 

Keywords: Indicator diagram; newton polyhedron; overlapping segment; p-adic sizes

 

ABSTRAK

Dalam makalah ini kami menganggarkan saiz p-adic pensifar sepunya polinomial-polinomial terbitan separa disekutukan dengan bentuk kubik dan gambar rajah penunjuknya mempunyai satu tembereng bertindih dengan menggunakan teknik polihedron Newton. Kami menunjukkan bahawa saiz p-adic bagi pensifar sepunya tersebut boleh didapati secara eksplisit pada tembereng bertindih dalam gambar rajah penunjuk disekutukan dengan polinomial terbabit.

 

 Kata kunci: Gambar rajah penunjuk; polihedron Newton; saiz p-adic; tembereng bertindih

 

 

RUJUKAN

 

Chan K.L. & Mohd. Atan K.A. Mohd. 1997. On the estimate to solutions of congruence equations associated with a quartic form. Journal of Physical Science 8: 21-34.

Heng S.H. & Mohd. Atan K.A. 1999. An estimation of exponential sums associated with a cubic form, Journal of Physical Science 10: 1-21.

Koblitz, N. 1977. p-Adic Numbers, p-Adic Analysis and Zeta-Function. New York: Springer-Verlag.

Mohd. Atan K.A. 1986a. Newton polyhedral method of determining p-adic orders of zeros common to two polynomials in Qp[x,y]. Pertanika 9(3): 375-380.

Mohd. Atan K. A. 1986b. Newton polyhedra and p-Adic estimates of zeros of polynomials in Ωp[x,y]. Pertanika 9(1): 51-56.

Mohd. Atan K.A. & Loxton J.H. 1986. Newton polyhedra and solutions of congruences. In Diophantine Analysis. Loxton J.H. and A. Van der Poorten (ed.). Cambridge: Cambridge University Press.

Sapar S.H. & Mohd Atan K.A. 2002. Estimate for the cardinality of the set of solution to congruence equations. Journal of Technology 36(C): 13-40.

Sapar S.H. & Mohd Atan K.A. 2007. Penganggaran saiz p-Adic pensifar sepunya terbitan separa polinomial berdarjah enam, Sains Malaysiana 36(1): 77-82.

 

*Pengarang untuk surat-menyurat; email: yaphongkeat@inspem.upm.edu.my

 

 

 

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