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763 lines
13 KiB
763 lines
13 KiB
#This file was created by <bruno> Sun Feb 16 14:19:06 1997
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#LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team
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\lyxformat 2.10
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\textclass article
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\begin_preamble
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\catcode`@=11 % @ ist ab jetzt ein gewoehnlicher Buchstabe
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\def\Res{\mathop{\operator@font Res}}
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\def\ll{\langle\!\langle}
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\def\gg{\rangle\!\rangle}
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\catcode`@=12 % @ ist ab jetzt wieder ein Sonderzeichen
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\end_preamble
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\language default
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\inputencoding latin1
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\fontscheme default
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\epsfig dvips
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\papersize a4paper
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\paperfontsize 12
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\baselinestretch 1.00
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\secnumdepth 3
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\tocdepth 3
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\paragraph_separation indent
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\quotes_language english
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\quotes_times 2
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\paperorientation portrait
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\papercolumns 0
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\papersides 1
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\paperpagestyle plain
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\layout LaTeX Title
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The diagonal of a rational function
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\layout Description
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Theorem:
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\layout Standard
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Let
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\begin_inset Formula \( M \)
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\end_inset
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be a torsion-free
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\begin_inset Formula \( R \)
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\end_inset
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-module, and
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\begin_inset Formula \( d>0 \)
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\end_inset
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.
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Let
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\begin_inset Formula
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\[
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f=\sum _{n_{1},...,n_{d}}a_{n_{1},...,n_{d}}\, x_{1}^{n_{1}}\cdots x_{d}^{n_{d}}\in M[[x_{1},\ldots x_{d}]]\]
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\end_inset
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be a rational function, i.
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e.
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there are
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\begin_inset Formula \( P\in M[x_{1},\ldots ,x_{d}] \)
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\end_inset
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and
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\begin_inset Formula \( Q\in R[x_{1},\ldots ,x_{d}] \)
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\end_inset
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with
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\begin_inset Formula \( Q(0,\ldots ,0)=1 \)
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\end_inset
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and
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\begin_inset Formula \( Q\cdot f=P \)
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\end_inset
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.
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Then the full diagonal of
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\begin_inset Formula \( f \)
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\end_inset
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\begin_inset Formula
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\[
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g=\sum ^{\infty }_{n=0}a_{n,\ldots ,n}\, x_{1}^{n}\]
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\end_inset
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is a D-finite element of
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\begin_inset Formula \( M[[x_{1}]] \)
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\end_inset
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, w.
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r.
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t.
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\begin_inset Formula \( R[x_{1}] \)
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\end_inset
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and
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\begin_inset Formula \( \{\partial _{x_{1}}\} \)
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\end_inset
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.
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\layout Description
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Proof:
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\layout Standard
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From the hypotheses,
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\begin_inset Formula \( M[[x_{1},\ldots ,x_{d}]] \)
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\end_inset
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is a torsion-free differential module over
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\begin_inset Formula \( R[x_{1},\ldots ,x_{d}] \)
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\end_inset
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w.
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r.
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t.
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the derivatives
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\begin_inset Formula \( \{\partial _{x_{1}},\ldots ,\partial _{x_{d}}\} \)
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\end_inset
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, and
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\begin_inset Formula \( f \)
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\end_inset
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is a D-finite element of
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\begin_inset Formula \( M[[x_{1},\ldots ,x_{d}]] \)
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\end_inset
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over
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\begin_inset Formula \( R[x_{1},\ldots ,x_{d}] \)
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\end_inset
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w.
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r.
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t.
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\begin_inset Formula \( \{\partial _{x_{1}},\ldots ,\partial _{x_{d}}\} \)
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\end_inset
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.
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Now apply the general diagonal theorem ([1], section 2.
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18)
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\begin_inset Formula \( d-1 \)
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\end_inset
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times.
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\layout Description
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Theorem:
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\layout Standard
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Let
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\begin_inset Formula \( R \)
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\end_inset
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be an integral domain of characteristic 0 and
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\begin_inset Formula \( M \)
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\end_inset
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simultaneously a torsion-free
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\begin_inset Formula \( R \)
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\end_inset
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-module and a commutative
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\begin_inset Formula \( R \)
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\end_inset
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-algebra without zero divisors.
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Let
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\begin_inset Formula
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\[
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f=\sum _{m,n\geq 0}a_{m,n}x^{m}y^{n}\in M[[x,y]]\]
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\end_inset
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be a rational function.
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Then the diagonal of
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\begin_inset Formula \( f \)
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\end_inset
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\begin_inset Formula
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\[
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g=\sum ^{\infty }_{n=0}a_{n,n}\, x^{n}\]
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\end_inset
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is algebraic over
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\begin_inset Formula \( R[x] \)
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\end_inset
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.
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\layout Description
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Motivation
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\protected_separator
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of
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\protected_separator
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proof:
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\layout Standard
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The usual proof ([2]) uses complex analysis and works only for
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\begin_inset Formula \( R=M=C \)
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\end_inset
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.
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The idea is to compute
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\begin_inset Formula
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\[
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g(x^{2})=\frac{1}{2\pi i}\oint _{|z|=1}f(xz,\frac{x}{z})\frac{dz}{z}\]
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\end_inset
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This integral, whose integrand is a rational function in
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\begin_inset Formula \( x \)
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\end_inset
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and
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\begin_inset Formula \( z \)
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\end_inset
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, is calculated using the residue theorem.
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Since
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\begin_inset Formula \( f(x,y) \)
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\end_inset
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is continuous at
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\begin_inset Formula \( (0,0) \)
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\end_inset
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, there is a
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\begin_inset Formula \( \delta >0 \)
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\end_inset
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such that
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\begin_inset Formula \( f(x,y)\neq \infty \)
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\end_inset
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for
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\begin_inset Formula \( |x|<\delta \)
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\end_inset
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,
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\begin_inset Formula \( |y|<\delta \)
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\end_inset
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.
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It follows that for all
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\begin_inset Formula \( \varepsilon >0 \)
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\end_inset
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and
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\begin_inset Formula \( |x|<\delta \varepsilon \)
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\end_inset
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all the poles of
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\begin_inset Formula \( f(xz,\frac{x}{z}) \)
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\end_inset
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are contained in
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\begin_inset Formula \( \{z:|z|<\varepsilon \}\cup \{z:|z|>\frac{1}{\varepsilon }\} \)
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\end_inset
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.
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Thus the poles of
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\begin_inset Formula \( f(xz,\frac{x}{z}) \)
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\end_inset
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, all algebraic functions of
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\begin_inset Formula \( x \)
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\end_inset
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-- let's call them
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\begin_inset Formula \( \zeta _{1}(x),\ldots \zeta _{s}(x) \)
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\end_inset
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--, can be divided up into those for which
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\begin_inset Formula \( |\zeta _{i}(x)|=O(|x|) \)
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\end_inset
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as
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\begin_inset Formula \( x\rightarrow 0 \)
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\end_inset
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and those for which
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\begin_inset Formula \( \frac{1}{|\zeta _{i}(x)|}=O(|x|) \)
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\end_inset
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as
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\begin_inset Formula \( x\rightarrow 0 \)
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\end_inset
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.
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It follows from the residue theorem that for
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\begin_inset Formula \( |x|<\delta \)
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\end_inset
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\begin_inset Formula
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\[
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g(x^{2})=\sum _{\zeta =0\vee \zeta =O(|x|)}\Res _{z=\zeta }\, f(xz,\frac{x}{z})\]
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\end_inset
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This is algebraic over
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\begin_inset Formula \( C(x) \)
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\end_inset
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.
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Hence
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\begin_inset Formula \( g(x) \)
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\end_inset
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is algebraic over
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\begin_inset Formula \( C(x^{1/2}) \)
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\end_inset
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, hence also algebraic over
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\begin_inset Formula \( C(x) \)
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\end_inset
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.
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\layout Description
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Proof:
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\layout Standard
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Let
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\begin_inset Formula
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\[
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h(x,z):=f(xz,\frac{x}{z})=\sum ^{\infty }_{m,n=0}a_{m,n}x^{m+n}z^{m-n}\in M[[xz,xz^{-1}]]\]
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\end_inset
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Then
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\begin_inset Formula \( g(x^{2}) \)
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\end_inset
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is the coefficient of
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\begin_inset Formula \( z^{0} \)
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\end_inset
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in
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\begin_inset Formula \( h(x,z) \)
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\end_inset
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.
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Let
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\begin_inset Formula \( N(x,z):=z^{d}Q(xz,\frac{x}{z}) \)
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\end_inset
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(with
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\begin_inset Formula \( d:=\max (\deg _{y}P,\deg _{y}Q) \)
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\end_inset
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) be
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\begin_inset Quotes eld
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\end_inset
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the denominator
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\begin_inset Quotes erd
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\end_inset
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of
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\begin_inset Formula \( h(x,z) \)
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\end_inset
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.
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We have
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\begin_inset Formula \( N(x,z)\in R[x,z] \)
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\end_inset
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and
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\begin_inset Formula \( N\neq 0 \)
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\end_inset
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(because
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\begin_inset Formula \( N(0,z)=z^{d} \)
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\end_inset
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).
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Let
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\begin_inset Formula \( K \)
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\end_inset
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be the quotient field of
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\begin_inset Formula \( R \)
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\end_inset
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.
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Thus
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\begin_inset Formula \( N(x,z)\in K[x][z]\setminus \{0\} \)
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\end_inset
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.
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\layout Standard
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It is well-known (see [3], p.
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64, or [4], chap.
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IV, §2, prop.
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8, or [5], chap.
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III, §1) that the splitting field of
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\begin_inset Formula \( N(x,z) \)
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\end_inset
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over
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\begin_inset Formula \( K(x) \)
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\end_inset
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can be embedded into a field
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\begin_inset Formula \( L((x^{1/r})) \)
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\end_inset
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, where
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\begin_inset Formula \( r \)
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\end_inset
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is a positive integer and
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\begin_inset Formula \( L \)
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\end_inset
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is a finite-algebraic extension field of
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\begin_inset Formula \( K \)
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\end_inset
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, i.
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e.
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a simple algebraic extension
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\begin_inset Formula \( L=K(\alpha )=K\alpha ^{0}+\cdots +K\alpha ^{u-1} \)
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\end_inset
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.
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\layout Standard
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\begin_inset Formula \( \widetilde{M}:=(R\setminus \{0\})^{-1}\cdot M \)
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\end_inset
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is a
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\begin_inset Formula \( K \)
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\end_inset
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-vector space and a commutative
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\begin_inset Formula \( K \)
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\end_inset
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-algebra without zero divisors.
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\begin_inset Formula \( \widehat{M}:=\widetilde{M}\alpha ^{0}+\cdots +\widetilde{M}\alpha ^{u-1} \)
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\end_inset
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is an
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\begin_inset Formula \( L \)
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\end_inset
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-vector space and a commutative
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\begin_inset Formula \( L \)
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\end_inset
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-algebra without zero divisors.
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\layout Standard
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\begin_inset Formula
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\begin{eqnarray*}
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\widehat{M}\ll x,z\gg & := & \widehat{M}[[x^{1/r}\cdot z,x^{1/r}\cdot z^{-1},x^{1/r}]][x^{-1/r}]\\
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& = & \left\{ \sum _{m,n}c_{m,n}x^{m/r}z^{n}:c_{m,n}\neq 0\Rightarrow |n|\leq m+O(1)\right\}
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\end{eqnarray*}
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\end_inset
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is an
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\begin_inset Formula \( L \)
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\end_inset
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-algebra which contains
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\begin_inset Formula \( \widehat{M}((x^{1/r})) \)
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\end_inset
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.
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\layout Standard
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Since
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\begin_inset Formula \( N(x,z) \)
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\end_inset
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splits into linear factors in
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\begin_inset Formula \( L((x^{1/r}))[z] \)
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\end_inset
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,
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\begin_inset Formula \( N(x,z)=l\prod ^{s}_{i=1}(z-\zeta _{i}(x))^{k_{i}} \)
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\end_inset
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, there exists a partial fraction decomposition of
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\begin_inset Formula \( h(x,z)=\frac{P(xz,\frac{x}{z})}{Q(xz,\frac{x}{z})}=\frac{z^{d}P(xz,\frac{x}{z})}{N(x,z)} \)
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\end_inset
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in
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\begin_inset Formula \( \widehat{M}\ll x,z\gg \)
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\end_inset
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:
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\layout Standard
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|
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\begin_inset Formula
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\[
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h(x,z)=\sum ^{l}_{j=0}P_{j}(x)z^{j}+\sum ^{s}_{i=1}\sum ^{k_{i}}_{k=1}\frac{P_{i,k}(x)}{(z-\zeta _{i}(x))^{k}}\]
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\end_inset
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with
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\begin_inset Formula \( P_{j}(x),P_{i,k}(x)\in \widehat{M}((x^{1/r})) \)
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\end_inset
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.
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\layout Standard
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|
Recall that we are looking for the coefficient of
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\begin_inset Formula \( z^{0} \)
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\end_inset
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in
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\begin_inset Formula \( h(x,z) \)
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\end_inset
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.
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We compute it separately for each summand.
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\layout Standard
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|
If
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\begin_inset Formula \( \zeta _{i}(x)=ax^{m/r}+... \)
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\end_inset
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with
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\begin_inset Formula \( a\in L\setminus \{0\} \)
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\end_inset
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,
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\begin_inset Formula \( m>0 \)
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\end_inset
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, or
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\begin_inset Formula \( \zeta _{i}(x)=0 \)
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\end_inset
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, we have
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\layout Standard
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|
|
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\begin_inset Formula
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\begin{eqnarray*}
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\frac{1}{(z-\zeta _{i}(x))^{k}} & = & \frac{1}{z^{k}}\cdot \frac{1}{\left( 1-\frac{\zeta _{i}(x)}{z}\right) ^{k}}\\
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& = & \frac{1}{z^{k}}\cdot \sum ^{\infty }_{j=0}{k-1+j\choose k-1}\left( \frac{\zeta _{i}(x)}{z}\right) ^{j}\\
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& = & \sum ^{\infty }_{j=0}{k-1+j\choose k-1}\frac{\zeta _{i}(x)^{j}}{z^{k+j}}
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\end{eqnarray*}
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\end_inset
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hence the coefficient of
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\begin_inset Formula \( z^{0} \)
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\end_inset
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in
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\begin_inset Formula \( \frac{P_{i,k}(x)}{(z-\zeta _{i}(x))^{k}} \)
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\end_inset
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is
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\begin_inset Formula \( 0 \)
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\end_inset
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.
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\layout Standard
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|
\cursor 59
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If
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\begin_inset Formula \( \zeta _{i}(x)=ax^{m/r}+... \)
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\end_inset
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with
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\begin_inset Formula \( a\in L\setminus \{0\} \)
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\end_inset
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,
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\begin_inset Formula \( m<0 \)
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\end_inset
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, we have
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\begin_inset Formula
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\[
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\frac{1}{(z-\zeta _{i}(x))^{k}}=\frac{1}{(-\zeta _{i}(x))^{k}}\cdot \frac{1}{\left( 1-\frac{z}{\zeta _{i}(x)}\right) ^{k}}=\frac{1}{(-\zeta _{i}(x))^{k}}\cdot \sum _{j=0}^{\infty }{k-1+j\choose k-1}\left( \frac{z}{\zeta _{i}(x)}\right) ^{j}\]
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\end_inset
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hence the coefficient of
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\begin_inset Formula \( z^{0} \)
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\end_inset
|
|
|
|
in
|
|
\begin_inset Formula \( \frac{P_{i,k}(x)}{(z-\zeta _{i}(x))^{k}} \)
|
|
\end_inset
|
|
|
|
is
|
|
\begin_inset Formula \( \frac{P_{i,k}(x)}{(-\zeta _{i}(x))^{k}} \)
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\layout Standard
|
|
|
|
The case
|
|
\begin_inset Formula \( \zeta _{i}(x)=ax^{m/r}+... \)
|
|
\end_inset
|
|
|
|
with
|
|
\begin_inset Formula \( a\in L\setminus \{0\} \)
|
|
\end_inset
|
|
|
|
,
|
|
\begin_inset Formula \( m=0 \)
|
|
\end_inset
|
|
|
|
, cannot occur, because it would imply
|
|
\begin_inset Formula \( 0=N(0,\zeta _{i}(0))=N(0,a)=a^{d}. \)
|
|
\end_inset
|
|
|
|
|
|
\layout Standard
|
|
|
|
Altogether we have
|
|
\begin_inset Formula
|
|
\[
|
|
g(x^{2})=[z^{0}]h(x,z)=P_{0}(x)+\sum _{\frac{1}{\zeta _{i}(x)}=o(x)}\sum ^{k_{i}}_{k=1}\frac{P_{i,k}(x)}{(-\zeta _{i}(x))^{k}}\in \widehat{M}((x^{1/r}))\]
|
|
|
|
\end_inset
|
|
|
|
|
|
\layout Standard
|
|
|
|
Since all
|
|
\begin_inset Formula \( \zeta _{i}(x) \)
|
|
\end_inset
|
|
|
|
(in
|
|
\begin_inset Formula \( L((x^{1/r})) \)
|
|
\end_inset
|
|
|
|
) and all
|
|
\begin_inset Formula \( P_{j}(x),P_{i,k}(x) \)
|
|
\end_inset
|
|
|
|
(in
|
|
\begin_inset Formula \( \widehat{M}((x^{1/r})) \)
|
|
\end_inset
|
|
|
|
) are algebraic over
|
|
\begin_inset Formula \( K(x) \)
|
|
\end_inset
|
|
|
|
, the same holds also for
|
|
\begin_inset Formula \( g(x^{2}) \)
|
|
\end_inset
|
|
|
|
.
|
|
Hence
|
|
\begin_inset Formula \( g(x) \)
|
|
\end_inset
|
|
|
|
is algebraic over
|
|
\begin_inset Formula \( K(x^{1/2}) \)
|
|
\end_inset
|
|
|
|
, hence also over
|
|
\begin_inset Formula \( K(x) \)
|
|
\end_inset
|
|
|
|
.
|
|
After clearing denominators, we finally conclude that
|
|
\begin_inset Formula \( g(x) \)
|
|
\end_inset
|
|
|
|
is algebraic over
|
|
\begin_inset Formula \( R[x] \)
|
|
\end_inset
|
|
|
|
.
|
|
|
|
\layout Bibliography
|
|
|
|
[1] Bruno Haible: D-finite power series in several variables.
|
|
|
|
\shape italic
|
|
Diploma thesis, University of Karlsruhe, June 1989.
|
|
|
|
\shape default
|
|
Sections 2.
|
|
18 and 2.
|
|
20.
|
|
|
|
\layout Bibliography
|
|
|
|
[2] M.
|
|
L.
|
|
J.
|
|
Hautus, D.
|
|
A.
|
|
Klarner: The diagonal of a double power series.
|
|
|
|
\shape italic
|
|
Duke Math.
|
|
J.
|
|
|
|
\shape default
|
|
|
|
\series bold
|
|
38
|
|
\series default
|
|
(1971), 229-235.
|
|
|
|
\layout Bibliography
|
|
|
|
[3] C.
|
|
Chevalley: Introduction to the theory of algebraic functions of one variable.
|
|
|
|
\shape italic
|
|
Mathematical Surveys VI.
|
|
American Mathematical Society.
|
|
|
|
\layout Bibliography
|
|
|
|
[4] Jean-Pierre Serre: Corps locaux.
|
|
|
|
\shape italic
|
|
Hermann.
|
|
Paris
|
|
\shape default
|
|
1968.
|
|
|
|
\layout Bibliography
|
|
|
|
[5] Martin Eichler: Introduction to the theory of algebraic numbers and
|
|
functions.
|
|
|
|
\shape italic
|
|
Academic Press.
|
|
New York, London
|
|
\shape default
|
|
1966.
|
|
|