## 数学代写|黎曼曲面代写Riemann surface代考|MAST30024

2023年2月1日

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## 数学代写|黎曼曲面代写Riemann surface代考|MOVING THE CYCLE OF INTEGRATION

We will now apply the procedure outlined in the previous sections to move the cycle of integration $\eta$ to obtain an analytic continuation.

Inductive hypothesis: Suppose that $\zeta_0$ is a point in $\mathbf{C}$, with a path $\rho_0$ from $|\zeta| \geq a$ to $\zeta_0$, of length $\leq M_0$, not meeting $S_{M_0}$. Suppose that $\eta_$ is a prochain with $(\partial-A) \eta_=0$, such that all points of $S u p p_{Z_}\left(\eta_\right)$ are beyond points of $S u p p_{z_}\left(\beta_\right)$ at distance $\leq M_0$, and such that there is an estimate $\mathbf{F}\left(\eta_n, \varepsilon n, C, \varepsilon\right)$ with $C$ and $\varepsilon$ uniform for all $\eta_n$. Suppose that $\zeta_0$ is not contained in $S u p p_{\mathbf{C}}\left(\eta_*\right)$, and that
$$f(\zeta)=\int_\eta \frac{b}{g-\zeta}$$
serves to define the analytic continuation of $f$ along the path $\rho_0$, for $\zeta$ near $\zeta_0$. Finally, assume that the path $\rho_0$ is piecewise linear, and that the chain $\eta$ is obtained by repeated applications of the procedure we are about to outline.
Fix a number $L$ and let $M=M_0+L$. Suppose $\rho:[0,1] \rightarrow \mathbf{C}$ is a line segment of length $L$, which does not meet $S_M$, and which begins at $\rho(0)=\zeta_0$. We would like to analytically continue $f(\zeta)$ along the segment $\rho$. Without loss of generality, we may make a rotation and assume that the segment points in the negative real direction, in other words $\rho(1)=\zeta_0-L$.

Choose a small number $\epsilon$ such that the disc $D\left(\zeta_0, 10 \epsilon\right)$ does not meet $\operatorname{Supp} p_{\mathbf{C}}\left(\eta_*\right)$, and such that the neighborhood $D(\rho, 10 \epsilon)$ (signifying the set of all points at distance less than $10 \epsilon$ from the segment $\rho$ ) does not meet $S_M$. Choose numbers $\sigma$ and $\delta$, small enough to meet the requirements made below. Let $L^{\prime}=L-\epsilon$. Let $\xi_0=\Re \zeta_0$.

Make a choice of flows as in $\S 3$, with reference to this angular error $\delta$, the length $L^{\prime}$ (which is slightly shorter than $L$ ), and the small number $\sigma$. Remember that a rotation has been made, so the flows should be chosen to go approximately in the negative real direction. In terms of a picture fixed from the start, the flows would go in some direction approximately equal to the direction of the line segment $\rho$. The flow $f^0$ must be fixed independently of which rotation is made, because it appears in the definition of the estimate $\mathbf{F}$, hence in the inductive hypothesis.

## 数学代写|黎曼曲面代写Riemann surface代考|BOUNDS ON MULTIPLICITIES

In this section we will use the assumption that the chain $\eta$ is obtained from $\beta$ by finitely many applications of the procedure outlined in $\S \S 4-8$, to prove that if $z$ is a generic point of $\Lambda(\ell)=\Lambda\left(\ell_1\right) \times \ldots \times \Lambda\left(\ell_n\right)$ then the multiplicity of $F H \varphi$ at $z$ is bounded by $C^n$. This was an ingredient in the previous section’s proof of (2.5.3).

First we give a general description of the chains which can arise from repeated applications of the procedures outlined in the previous section.

Suppose $A$ and $B$ are subsets of $\mathbf{R}^a$ and $\mathbf{R}^b$ respectively. A continuous $\operatorname{map} f: A \rightarrow B$ is piecewise polynomial if there is a finite decomposition $A=\bigcup U_\alpha$ and if there are polynomial maps $P_\alpha$ from $\mathbf{R}^a$ to $\mathbf{R}^b$ such that $\left.f\right|{U\alpha}=P_\alpha$. It follows from continuity that the boundaries between pieces are algebraically defined. The degree of $f$ is the largest of the degrees of the component polynomials $P_{\alpha, i}, i=1, \ldots, b$. If $s_i$ and $t_j$ are coordinates in $\mathbf{R}^a$ and $\mathbf{R}^b$ respectively, then the degree of $f_j$ in the variable $s_i$ is the largest of the degrees of the polynomials $P_{\alpha, j}$ in the variable $s_i$. The size of $f$ is the largest of the following numbers: the number of pieces $U_\alpha$, and the suprema of the derivatives $\sup {U_a}\left|\partial P{\alpha, i} / \partial x_j\right|$.

If $f: A \rightarrow B$ and $g: B \rightarrow C$ are piecewise polynomial maps, then $g \circ f$ : $A \rightarrow C$ is a piecewise polynomial map. Furthermore, the degree of $g \circ f$ is less than or equal to the product of the degrees of $f$ and $g$. The number of pieces into which the map $g \circ f$ is decomposed is less than or equal to the product of the number of pieces for $g$ and the number of pieces for $f$. The supremum of the partial derivatives of $g \circ f$ is less than or equal to $\operatorname{dim}(B)$ times the product of the suprema of the partial derivatives of $f$ and $g$, by the chain rule.

We now define a type of piecewise polynomial map which will arise in describing the chains that can occur. Suppose we have a graph organized as a tree, with some edges marked and some not, beginning with $m$ vertices along the top and ending with $n$ vertices along the bottom. Suppose that for each marked edge we have a map $f(e, x, t): Z \times[0,1] \rightarrow Z$, and suppose that for each unmarked edge $e$ we have a map $f(e, x): Z \rightarrow Z$. Suppose that there are $N$ marked edges. Then we get a map
$$\Phi: Z^m \times[0,1]^N \rightarrow Z^n$$
defined as follows (it is similar to our usual construction seen first in §5). Fix $z \in Z^m$ and $s \in[0,1]^N$.

# 黎曼曲面代写

## 数学代写|黎曼曲面代写Riemann surface代考|BOUNDS ON MULTIPLICITIES

$\Lambda(\ell)=\Lambda\left(\ell_1\right) \times \ldots \times \Lambda\left(\ell_n\right)$ 然后的多重性 $F H \varphi$ 在 $z$ 受限于 $C^n$. 这是上一节 (2.5.3) 证明中的一个组成部分。

$$\Phi: Z^m \times[0,1]^N \rightarrow Z^n$$
$$z \in Z^m s \in[0,1]^N$$

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