关于（De）齐次Gröbner基
On (De)homogenized Gröbner Bases
摘要：
设$K$是一个域，$R=\oplus_{p\in\mathbb{N}}R_p$是一个$\mathbb{N}$ -分次$K$ -代数，它有一个SM $K$ -基（即斜乘法$K$ -基），使得$R$保持Gröbner基理论.证明了多项式代数$R$中的Gröbner基集与多项式代数$R[t]$中的dh-闭齐次Gröbner基集之间存在一一对应关系，并证明了若分别用自由代数$K< X_1,...,X_n>$和自由代数$K< X_1,...,X_n,T>$代替$R$和$R[t]$，同样的结果也成立.证明了$R[t]$和$K< X_1,...,X_n, T>$中的dh-闭分次理想可以由dh-闭齐次Gröbner基实现.后一个结果确实告诉我们，由dh-齐次Gröbner基定义的代数可以通过更简单的代数有效地研究为Rees代数，如（[7]，[8]）所证明的。
英文摘要：
Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded $K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis) such that $R$ holds a Gröbner basis theory. It is proved that there is a one-to-one correspondence between the set of Gröbner bases in $R$ and the set of dh-closed homogeneous Gröbner bases in the polynomial algebra $R[t]$; and that the similar result holds true if $R$ and $R[t]$ are replaced respectively by the free algebra $K< X_1,...,X_n>$ and the free algebra $K< X_1,...,X_n,T>$. Moreover, it is shown that dh-closed graded ideals in $R[t]$ and $K< X_1,...,X_n, T>$ can be realized by dh-closed homogeneous Gröbner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Gröbner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]).
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