# 应用数学入门 Introduction to Applied Mathematics MAT00003C

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Let $R=K\left[x_{1}, \ldots, x_{n}\right]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $R$ generated by a finite set of monomials $\left{x^{v_{1}}, \ldots, x^{v_{q}}\right}$. As usual we use $x^{a}$ as an abbreviation for $x_{1}^{a_{1}} \cdots x_{n}^{a_{n}}$, where $a=\left(a_{1}, \ldots, a_{n}\right)$ is in $\mathbb{N}^{n}$. The three central objects of study here are the following blowup algebras: (a) the Rees algebra
$$R[I t]:=R \oplus I t \oplus \cdots \oplus I^{i} t^{i} \oplus \cdots \subset R[t],$$
where $t$ is a new variable, (b) the associated graded ring
$$\operatorname{gr}{I}(R):=R / I \oplus I / I^{2} \oplus \cdots \oplus I^{i} / I^{i+1} \oplus \cdots \simeq R[I t] \otimes{R}(R / I),$$

with multiplication
$$\left(a+I^{i+1}\right)\left(b+I^{j+1}\right)=a b+I^{i+j+1} \quad\left(a \in I^{i}, b \in I^{j}\right),$$
and (c) the symbolic Rees algebra
$$R_{s}(I):=R+I^{(1)} t+I^{(2)} t^{2}+\cdots+I^{(t)} t^{i}+\cdots \subset R[t],$$
where $I^{(i)}$ is the ith symbolic power of $I$.

## MAT00003C COURSE NOTES ：

Let $R=k\left[x_{1}, \ldots, x_{n}\right]$ be a polynomial ring over a field $k$. Suppose $M=x_{1} a_{1} \ldots x_{n} a_{n}$ is a monomial in $R$. Then we define the polarization of $M$ to be the square-free monomial
$$\mathscr{P}(M)=x_{1,1} x_{1,2} \ldots x_{1, a_{1}} x_{2,1} \ldots x_{2, a_{2}} \ldots x_{n, 1} \ldots x_{n, a_{n}}$$
in the polynomial ring $S=k\left[x_{i, j} \mid 1 \leq i \leq n, 1 \leq j \leq a_{i}\right]$.
If $I$ is an ideal of $R$ generated by monomials $M_{1}, \ldots, M_{q}$, then the polarization of $I$ is defined as:
$$P(I)=\left(P\left(M_{1}\right), \ldots, P\left(M_{q}\right)\right)$$