OnHilbert-KunzFunctionofBinomialHypersurfaces

• Main Author: 謝信鴻
• Other Authors: 洪有情
• Format: Others
• Language: en_US
• Online Access: http://ndltd.ncl.edu.tw/handle/07484736150486094939

碩士 === 國立臺灣師範大學 === 數學研究所 === 85 === Let (R,m) be a complete local noetherian Z/(p)-algebra, and m[n] the Pn-th Frobenius power of m, i.e. the ideal generated by the Pn -th powers of the elements of m . Let en(R) be the length of R/m[n]. The function n|→en (R) is called the Hilbert-Kunz function of R. Let F be a field of characteristic p ＞ 0 and S＝F〔|S1,...,Xr,Y1,...,Ys,Z1,...,Zt|〕. In this article, by making use of Froebner basis, we determine the Hilbert-Kunz function of binomial hypersurfaces of the form: f：＝X1(a1...Xr(ar))Y1(b1)...Ys(bs)＋Y1(c1)...Ys(cs)Z1(d1))...Zt(dr) with bk≧ck for each k. LetR＝S/<f>，. In Section 2, we discuss the case r≧1 and the discussion for r＝0 is in section 3. In Proposition 2.7 and 3.7, we explicitly determine the Hilbert-Kunz function, and in Theorem 2.8 and 3.8, we show that for n>>0, where λ is a rational number and fk(n) is an eventually periodic function of n for each k. For the general binomial hypersufaces, it is more complicated to explicitly determine the Hibert-Kunz function.