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# 数学代考|微分方程代考differential equation作业代写|Formal flows and embedding theorem

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## 数学代考|微分方程代考differantial equation作业代写|Formal vector fields and formal morphisms

For convenience, we will always assume that all Taylor series are centered at the origin.

A formal (Taylor) series at the origin in $\mathbb{C}^{n}$ is an expression
$$f=\sum_{\alpha} c_{\alpha} x^{\alpha}, \quad \alpha \in \mathbb{Z}{+}^{n}, \quad c{\alpha} \in \mathbb{C} .$$
The minimal degree $|\alpha|$ corresponding to a nonzero coefficient $c_{\alpha}$, is called the order of $f$.

The set of all formal series is denoted by $\mathbb{C}[[x]]=\mathbb{C}\left[\left[x_{1}, \ldots, x_{n}\right]\right]$. It is a commutative infinite-dimensional algebra over $\mathbb{C}$ which contains as a proper subset the algebra of germs of holomorphic functions, isomorphic to the algebra $\mathbb{C}\left{x_{1}, \ldots, x_{n}\right}$ of converging series.

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|Inverse function theorem

For future purposes we will need the formal inverse function theorem.

Let $H$ be a formal map with the linearization matrix $A=$ $\left(\frac{\partial H}{\partial x}\right)(0)$ which is nondegenerate. Then $H$ is invertible in Diff $\left[\left[\mathbb{C}^{n}, 0\right]\right]$.
If $A=E$ is the identity matrix and $H=\left(h_{1}, \ldots, h_{n}\right), h_{i}(x)=x_{i}+$ $v_{i}(x) \bmod \mathrm{m}^{k+1}$, where $v_{i}$ are homogeneous polynomials of degree $k \geqslant 2$, then the formal inverse map $H^{-1}=\left(h_{1}^{\prime}, \ldots, h_{n}^{\prime}\right)$ has the components $h_{i}^{\prime}(x)=$ $x_{i}-v_{i}(x) \bmod \mathfrak{m}^{k+1}$.

Clearly, it the first assertion of the Theorem follows from the second assertion applied to the formal map $A^{-1} H$.

The following definition will play important role throughout this section.

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|Integration and formal flow of formal vector fields

Consider an (autonomous) formal ordinary differential equation
$$\dot{x}=F(x), \quad F=\left(F_{1}, \ldots, F_{n}\right) \in \mathcal{D}\left[\left[\mathbb{C}^{n}, 0\right]\right] \simeq \mathbb{C}[[x]]^{n}$$
with a formal right hand side part $F$. Since evaluation of a formal series at any point other than the origin makes no sense, the “standard” definition of solutions can at best be applied to constructing a solution with the initial condition $x(0)=0$. Yet in the most interesting case when $F(0)=0$, this solution is trivial, $x(t) \equiv 0$.

The alternative, suggested by Remark $1.20$, is to define a one-parametric subgroup of formal automorphisms $\left{\mathbf{H}^{t}: t \in \mathbb{C}\right} \subset \operatorname{Diff}\left[\left[\mathbb{C}^{n}, 0\right]\right]$ satisfying the condition
$$\mathbf{H}^{t} \circ \mathbf{H}^{s}=\mathbf{H}^{t+s} \quad \forall t, s \in \mathbb{C}, \quad \mathbf{H}^{0}=\mathbf{E} .$$
The corresponding condition for formal maps takes the form $H^{t}\left(H^{s}(x)\right)=$ $H^{s}\left(H^{t}(x)\right)=H^{t+s}(x)$.

This subgroup is said to be holomorphic, if all finite truncations $j^{k} \mathbf{H}^{t}$ depend holomorphically on $t$. For a holomorphic subgroup the derivative
$$\mathbf{F}=\left.\frac{d \mathbf{H}^{t}}{d t}\right|{t=0}=\lim {t \rightarrow 0} t^{-1}\left(\mathbf{H}^{t}-\mathbf{E}\right): \mathbb{C}[[x]] \rightarrow \mathbb{C}[[x]] .$$

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|FORMAL VECTOR FIELDS AND FORMAL MORPHISMS

$$f=\sum_{\alpha} c_{\alpha} x^{\alpha}, \quad \alpha \in \mathbb{Z} {+}^{n}, \quad c {\alpha } \in \mathbb{C} 。$$

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|INVERSE FUNCTION THEOREM

Then $H$ is invertible in Diff $\left[\left[\mathbb{C}^{n}, 0\right]\right]$.
If $A=E$ is the identity matrix and $H=\left(h_{1}, \ldots, h_{n}\right), h_{i}(x)=x_{i}+$ $v_{i}(x) \bmod \mathrm{m}^{k+1}$, where $v_{i}$ are homogeneous polynomials of degree $k \geqslant 2$, then the formal inverse map $H^{-1}=\left(h_{1}^{\prime}, \ldots, h_{n}^{\prime}\right)$ has the components $h_{i}^{\prime}(x)=$ $x_{i}-v_{i}(x) \bmod \mathfrak{m}^{k+1}$.

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|INTEGRATION AND FORMAL FLOW OF FORMAL VECTOR FIELDS

X˙=F(X),F=(F1,…,Fn)∈D[[Cn,0]]≃C[[X]]n

Remark 建议的替代方案1.20, 是定义形式自同构的单参数子群\left{\mathbf{H}^{t}: t \in \mathbb{C}\right} \subset \operatorname{Diff}\left[\left[\mathbb{C}^{n}, 0\right ]\对]\left{\mathbf{H}^{t}: t \in \mathbb{C}\right} \subset \operatorname{Diff}\left[\left[\mathbb{C}^{n}, 0\right ]\对]满足条件
H吨∘Hs=H吨+s∀吨,s∈C,H0=和.

$$\mathbf{F}=\left.\frac{d \mathbf{H}^{t}}{dt}\right| {t=0}=\lim {t \rightarrow 0} t^{-1}\left\mathbf{H}^{t}-\mathbf{E}\right\mathbf{H}^{t}-\mathbf{E}\right: \mathbb{C}[X] \rightarrow \mathbb{C}[X] .$$

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