1 #This file was created by <bruno> Sun Feb 16 14:19:06 1997
2 #LyX 0.10 (C) 1995 1996 Matthias Ettrich and the LyX Team
6 \catcode`@=11 % @ ist ab jetzt ein gewoehnlicher Buchstabe
7 \def\Res{\mathop{\operator@font Res}}
8 \def\ll{\langle\!\langle}
9 \def\gg{\rangle\!\rangle}
10 \catcode`@=12 % @ ist ab jetzt wieder ein Sonderzeichen
22 \paragraph_separation indent
23 \quotes_language english
25 \paperorientation portrait
32 The diagonal of a rational function
39 \begin_inset Formula \( M \)
43 \begin_inset Formula \( R \)
47 \begin_inset Formula \( d>0 \)
54 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}]]\]
58 be a rational function, i.
61 \begin_inset Formula \( P\in M[x_{1},\ldots ,x_{d}] \)
65 \begin_inset Formula \( Q\in R[x_{1},\ldots ,x_{d}] \)
69 \begin_inset Formula \( Q(0,\ldots ,0)=1 \)
73 \begin_inset Formula \( Q\cdot f=P \)
77 Then the full diagonal of
78 \begin_inset Formula \( f \)
84 g=\sum ^{\infty }_{n=0}a_{n,\ldots ,n}\, x_{1}^{n}\]
88 is a D-finite element of
89 \begin_inset Formula \( M[[x_{1}]] \)
96 \begin_inset Formula \( R[x_{1}] \)
100 \begin_inset Formula \( \{\partial _{x_{1}}\} \)
111 \begin_inset Formula \( M[[x_{1},\ldots ,x_{d}]] \)
114 is a torsion-free differential module over
115 \begin_inset Formula \( R[x_{1},\ldots ,x_{d}] \)
122 \begin_inset Formula \( \{\partial _{x_{1}},\ldots ,\partial _{x_{d}}\} \)
126 \begin_inset Formula \( f \)
129 is a D-finite element of
130 \begin_inset Formula \( M[[x_{1},\ldots ,x_{d}]] \)
134 \begin_inset Formula \( R[x_{1},\ldots ,x_{d}] \)
141 \begin_inset Formula \( \{\partial _{x_{1}},\ldots ,\partial _{x_{d}}\} \)
145 Now apply the general diagonal theorem ([1], section 2.
147 \begin_inset Formula \( d-1 \)
158 \begin_inset Formula \( R \)
161 be an integral domain of characteristic 0 and
162 \begin_inset Formula \( M \)
165 simultaneously a torsion-free
166 \begin_inset Formula \( R \)
169 -module and a commutative
170 \begin_inset Formula \( R \)
173 -algebra without zero divisors.
177 f=\sum _{m,n\geq 0}a_{m,n}x^{m}y^{n}\in M[[x,y]]\]
181 be a rational function.
183 \begin_inset Formula \( f \)
189 g=\sum ^{\infty }_{n=0}a_{n,n}\, x^{n}\]
194 \begin_inset Formula \( R[x] \)
208 The usual proof ([2]) uses complex analysis and works only for
209 \begin_inset Formula \( R=M=C \)
213 The idea is to compute
216 g(x^{2})=\frac{1}{2\pi i}\oint _{|z|=1}f(xz,\frac{x}{z})\frac{dz}{z}\]
220 This integral, whose integrand is a rational function in
221 \begin_inset Formula \( x \)
225 \begin_inset Formula \( z \)
228 , is calculated using the residue theorem.
230 \begin_inset Formula \( f(x,y) \)
234 \begin_inset Formula \( (0,0) \)
238 \begin_inset Formula \( \delta >0 \)
242 \begin_inset Formula \( f(x,y)\neq \infty \)
246 \begin_inset Formula \( |x|<\delta \)
250 \begin_inset Formula \( |y|<\delta \)
254 It follows that for all
255 \begin_inset Formula \( \varepsilon >0 \)
259 \begin_inset Formula \( |x|<\delta \varepsilon \)
263 \begin_inset Formula \( f(xz,\frac{x}{z}) \)
267 \begin_inset Formula \( \{z:|z|<\varepsilon \}\cup \{z:|z|>\frac{1}{\varepsilon }\} \)
272 \begin_inset Formula \( f(xz,\frac{x}{z}) \)
275 , all algebraic functions of
276 \begin_inset Formula \( x \)
280 \begin_inset Formula \( \zeta _{1}(x),\ldots \zeta _{s}(x) \)
283 --, can be divided up into those for which
284 \begin_inset Formula \( |\zeta _{i}(x)|=O(|x|) \)
288 \begin_inset Formula \( x\rightarrow 0 \)
292 \begin_inset Formula \( \frac{1}{|\zeta _{i}(x)|}=O(|x|) \)
296 \begin_inset Formula \( x\rightarrow 0 \)
300 It follows from the residue theorem that for
301 \begin_inset Formula \( |x|<\delta \)
307 g(x^{2})=\sum _{\zeta =0\vee \zeta =O(|x|)}\Res _{z=\zeta }\, f(xz,\frac{x}{z})\]
311 This is algebraic over
312 \begin_inset Formula \( C(x) \)
317 \begin_inset Formula \( g(x) \)
321 \begin_inset Formula \( C(x^{1/2}) \)
324 , hence also algebraic over
325 \begin_inset Formula \( C(x) \)
338 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}]]\]
343 \begin_inset Formula \( g(x^{2}) \)
346 is the coefficient of
347 \begin_inset Formula \( z^{0} \)
351 \begin_inset Formula \( h(x,z) \)
356 \begin_inset Formula \( N(x,z):=z^{d}Q(xz,\frac{x}{z}) \)
360 \begin_inset Formula \( d:=\max (\deg _{y}P,\deg _{y}Q) \)
364 \begin_inset Quotes eld
368 \begin_inset Quotes erd
372 \begin_inset Formula \( h(x,z) \)
377 \begin_inset Formula \( N(x,z)\in R[x,z] \)
381 \begin_inset Formula \( N\neq 0 \)
385 \begin_inset Formula \( N(0,z)=z^{d} \)
390 \begin_inset Formula \( K \)
393 be the quotient field of
394 \begin_inset Formula \( R \)
399 \begin_inset Formula \( N(x,z)\in K[x][z]\setminus \{0\} \)
406 It is well-known (see [3], p.
410 III, §1) that the splitting field of
411 \begin_inset Formula \( N(x,z) \)
415 \begin_inset Formula \( K(x) \)
418 can be embedded into a field
419 \begin_inset Formula \( L((x^{1/r})) \)
423 \begin_inset Formula \( r \)
426 is a positive integer and
427 \begin_inset Formula \( L \)
430 is a finite-algebraic extension field of
431 \begin_inset Formula \( K \)
436 a simple algebraic extension
437 \begin_inset Formula \( L=K(\alpha )=K\alpha ^{0}+\cdots +K\alpha ^{u-1} \)
445 \begin_inset Formula \( \widetilde{M}:=(R\setminus \{0\})^{-1}\cdot M \)
449 \begin_inset Formula \( K \)
452 -vector space and a commutative
453 \begin_inset Formula \( K \)
456 -algebra without zero divisors.
458 \begin_inset Formula \( \widehat{M}:=\widetilde{M}\alpha ^{0}+\cdots +\widetilde{M}\alpha ^{u-1} \)
462 \begin_inset Formula \( L \)
465 -vector space and a commutative
466 \begin_inset Formula \( L \)
469 -algebra without zero divisors.
476 \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}]\\
477 & = & \left\{ \sum _{m,n}c_{m,n}x^{m/r}z^{n}:c_{m,n}\neq 0\Rightarrow |n|\leq m+O(1)\right\}
483 \begin_inset Formula \( L \)
486 -algebra which contains
487 \begin_inset Formula \( \widehat{M}((x^{1/r})) \)
495 \begin_inset Formula \( N(x,z) \)
498 splits into linear factors in
499 \begin_inset Formula \( L((x^{1/r}))[z] \)
503 \begin_inset Formula \( N(x,z)=l\prod ^{s}_{i=1}(z-\zeta _{i}(x))^{k_{i}} \)
506 , there exists a partial fraction decomposition of
507 \begin_inset Formula \( h(x,z)=\frac{P(xz,\frac{x}{z})}{Q(xz,\frac{x}{z})}=\frac{z^{d}P(xz,\frac{x}{z})}{N(x,z)} \)
511 \begin_inset Formula \( \widehat{M}\ll x,z\gg \)
520 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}}\]
525 \begin_inset Formula \( P_{j}(x),P_{i,k}(x)\in \widehat{M}((x^{1/r})) \)
532 Recall that we are looking for the coefficient of
533 \begin_inset Formula \( z^{0} \)
537 \begin_inset Formula \( h(x,z) \)
541 We compute it separately for each summand.
546 \begin_inset Formula \( \zeta _{i}(x)=ax^{m/r}+... \)
550 \begin_inset Formula \( a\in L\setminus \{0\} \)
554 \begin_inset Formula \( m>0 \)
558 \begin_inset Formula \( \zeta _{i}(x)=0 \)
567 \frac{1}{(z-\zeta _{i}(x))^{k}} & = & \frac{1}{z^{k}}\cdot \frac{1}{\left( 1-\frac{\zeta _{i}(x)}{z}\right) ^{k}}\\
568 & = & \frac{1}{z^{k}}\cdot \sum ^{\infty }_{j=0}{k-1+j\choose k-1}\left( \frac{\zeta _{i}(x)}{z}\right) ^{j}\\
569 & = & \sum ^{\infty }_{j=0}{k-1+j\choose k-1}\frac{\zeta _{i}(x)^{j}}{z^{k+j}}
574 hence the coefficient of
575 \begin_inset Formula \( z^{0} \)
579 \begin_inset Formula \( \frac{P_{i,k}(x)}{(z-\zeta _{i}(x))^{k}} \)
583 \begin_inset Formula \( 0 \)
591 \begin_inset Formula \( \zeta _{i}(x)=ax^{m/r}+... \)
595 \begin_inset Formula \( a\in L\setminus \{0\} \)
599 \begin_inset Formula \( m<0 \)
605 \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}\]
609 hence the coefficient of
610 \begin_inset Formula \( z^{0} \)
614 \begin_inset Formula \( \frac{P_{i,k}(x)}{(z-\zeta _{i}(x))^{k}} \)
618 \begin_inset Formula \( \frac{P_{i,k}(x)}{(-\zeta _{i}(x))^{k}} \)
626 \begin_inset Formula \( \zeta _{i}(x)=ax^{m/r}+... \)
630 \begin_inset Formula \( a\in L\setminus \{0\} \)
634 \begin_inset Formula \( m=0 \)
637 , cannot occur, because it would imply
638 \begin_inset Formula \( 0=N(0,\zeta _{i}(0))=N(0,a)=a^{d}. \)
647 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}))\]
655 \begin_inset Formula \( \zeta _{i}(x) \)
659 \begin_inset Formula \( L((x^{1/r})) \)
663 \begin_inset Formula \( P_{j}(x),P_{i,k}(x) \)
667 \begin_inset Formula \( \widehat{M}((x^{1/r})) \)
671 \begin_inset Formula \( K(x) \)
674 , the same holds also for
675 \begin_inset Formula \( g(x^{2}) \)
680 \begin_inset Formula \( g(x) \)
684 \begin_inset Formula \( K(x^{1/2}) \)
688 \begin_inset Formula \( K(x) \)
692 After clearing denominators, we finally conclude that
693 \begin_inset Formula \( g(x) \)
697 \begin_inset Formula \( R[x] \)
704 [1] Bruno Haible: D-finite power series in several variables.
707 Diploma thesis, University of Karlsruhe, June 1989.
721 Klarner: The diagonal of a double power series.
737 Chevalley: Introduction to the theory of algebraic functions of one variable.
740 Mathematical Surveys VI.
741 American Mathematical Society.
745 [4] Jean-Pierre Serre: Corps locaux.
755 [5] Martin Eichler: Introduction to the theory of algebraic numbers and