一些三項高次曲面的希爾伯特方程式

Abstract

這篇論文主要是利用Gröbner basis去判斷一些限制條件下的高次曲面的希爾伯特方程式，經過化簡我們得到的結論跟以往其他論文所做的結果一致。In this paper, by making use of Gr\"{o}bner basis, we determine the Hilbert-Kunz function of some trinomial hypersurfaces of the form \[f:=X^{a}Y^{b}+Y^{c}Z^{d}+Z^{e}\] with $0<a\leq b\leq c$, which is \[n\longmapsto \lambda p^{2n}+f_1(n)p^n+f_0(n)\] for $n\gg 0$, where $\lambda=\left[ \dps \sum_{k=1}^2(-1)^{k+1}S_k(a,b)\frac{e}{u^k}\right]$ and $f_k(n)$ is an eventually periodic function of $n$ for each $k$.