臺灣博碩士論文加值系統

在這篇文章中，藉著使用Groebner基底，我們要決定形如
f:= X1^a1...Xr^arY1^b1....Ys^bs + Y1^c1...Ys^csZ1^d1...Zt^dt
的二項式超曲面的Hilbert-Kunz 函數，並且觀察其中的特性。

Contents
Introduction 1
1 Preliminary 3
2 The First Form : X^a Y^b + Y^c Z^d 9
3 The Second Form : Y^b + Y^c Z^d 36
References 52

References
[1] Adams, W. W. and Loustaunau, P. : An Introduction to Gr¨ obner Bases. American Math-ematical
Society (1994)
[2] Buchweitz, R.O. and Chen, Q. : Hilbert-Kunz functions of cubic curves and surfaces.
Journal of Algebra 197, 246—267 (1997)
[3] Chiang, L. and Hung, Y.C. : On Hilbert-Kunz function and representation ring. Bull. Inst.
Math. Acad. Sinica Vol26, No1, 1—32 (1998)
[4] Chiang, L. and Hung, Y.C. : On Hilbert-Kunz Functions of Some Hypersurfaces. Journal
of Algebra 199, 499—527 (1998)
[5] Conca, A. : Hilbert-Kunz function of monomial ideals and binomial hypersurfaces.
Manuscripta Math. 90, 287—300 (1996)
[6] Cox, D., Little, J. and O’Shea, D. : Idaels, Varieties, and Algorithms. Springer-Verlag New
York, Inc. (1992)
[7] Han, C. and Monsky, P. : Some surprising Hilbert-Kunz functions. Math. Z. 214, 119—135
(1993)
[8] Hsieh, H. H. : On Hilbert-Kunz Function of Binomial Hypersurfaces. Master’s Thesis,
National Taiwan Normal University (1997)
[9] Kunz, E. : Characterizations of regular local rings of characteristic p. Am. J. Math. 41,
772—784 (1969)
[10] Kunz, E. : On Noetherian rings of characteristic p. Am. J. Math. 98, 999—1013 (1976)
[11] Monsky, P. : The Hilbert-Kunz function. Math. Ann. 263, 43—49 (1983)
[12] Monsky, P. : The Hilbert-Kunz functions of a characteristic 2 cubic. Journal of Algebra
197, 268—277 (1997)
[13] Pauer, F. and Pfeifhofer, M. : The Theory of Gr¨obner Bases. L’Enseignement Mathematique,
t.34, p.215—232 (1988)
[14] Wolmer, V. V. and Eisenbud, D. : Computational methods in commutative algebra and
algebraic geometry. Springer-Verlag Berlin Heidelberg New York (1998)