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# 数学代写|交换代数代写Commutative Algebra代考|Rings Associated to Monomial Orderings

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## 数学代写|交换代数代写Commutative Algebra代考|Rings Associated to Monomial Orderings

In this section we show that non-global monomial orderings lead to new rings which are localizations of the polynomial ring. This fact has far-reaching computational consequences. For example, choosing a local ordering, we can, basically, do the same calculations in the localization of a polynomial ring as with a global ordering in the polynomial ring itself. In particular, we can effectively compute in $K\left[x_1, \ldots, x_n\right]_{\left\langle x_1, \ldots, x_k\right\rangle}$ for $k \leq n$ (by Lemma 1.5.2 (3) and Example 1.5.3).

Let $>$ be a monomial ordering on the set of monomials Mon $\left(x_1, \ldots, x_n\right)=$ $\left{x^\alpha \mid \alpha \in \mathbb{N}^n\right}$, and $K[x]=K\left[x_1, \ldots, x_n\right]$ the polynomial ring in $n$ variables over a field $K$. Then the leading monomial function LM has the following properties for polynomials $f, g \in K[x] \backslash{0}$ :
(1) $\operatorname{LM}(g f)=\operatorname{LM}(g) \operatorname{LM}(f)$
(2) $\operatorname{LM}(g+f) \leq \max {\operatorname{LM}(g), \operatorname{LM}(f)}$ with equality if and only if the leading terms of $f$ and $g$ do not cancel.
In particular, it follows that
$$S_{>}:={u \in K[x] \backslash{0} \mid \operatorname{LM}(u)=1}$$
is a multiplicatively closed set.

## 数学代写|交换代数代写Commutative Algebra代考|Normal Forms and Standard Bases

In this section we define standard bases, respectively Gröbner bases, of an ideal $I \subset K[x]_{>}$as a set of polynomials of $I$ such that their leading monomials generate the leading ideal $L(I)$. The next section gives an algorithm to compute standard bases. For global orderings this is Buchberger’s algorithm, which is a generalization of the Gaussian elimination algorithm and the Euclidean algorithm. For local orderings it is Mora’s tangent cone algorithm, which itself is a variant of Buchberger’s algorithm. The general case is a variation of Mora’s algorithm, which is due to the authors and implemented in SiNGULAR since 1990.

The leading ideal $L(I)$ contains a lot of information about the ideal $I$, which often can be computed purely combinatorially from $L(I)$, because the leading ideal is generated by monomials. Standard bases have turned out to be the fundamental tool for computations with ideals and modules. The idea of standard bases is already contained in the work of Gordan [93]. Later, monomial orderings were used by Macaulay [157] and Gröbner [117] to study Hilbert functions of graded ideals, and, more generally, to find bases of zerodimensional factor rings. The notion of a standard basis was introduced later, independently, by Hironaka [123], Grauert [98] (for special local orderings) and Buchberger [32] (for global orderings).

In the following, special emphasis is made to axiomatically characterize normal forms, respectively weak normal forms, which play an important role in the standard basis algorithm. They generalize division with remainder to the case of ideals, respectively finite sets of polvnomials.

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