一些三項高次曲面的希爾伯特方程式
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2006
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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 $k$.
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 $k$.
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希爾伯特方程式, Gröbner基底, Gr\"{o}bner basis, Hilbert-Kunz function