From annick.valibouze@upmc.fr Tue Apr 11 10:10:40 2006 Return-Path: Delivered-To: online.fr-avbumpc@free.fr Received: (qmail 26918 invoked from network); 11 Apr 2006 08:11:51 -0000 Received: from shiva.jussieu.fr (134.157.0.129) by mrelay3-2.free.fr with SMTP; 11 Apr 2006 08:11:51 -0000 Received: from smtp3-g19.free.fr (smtp3-g19.free.fr [212.27.42.29]) by shiva.jussieu.fr (8.13.6/jtpda-5.4) with ESMTP id k3B8Bmga021136 for ; Tue, 11 Apr 2006 10:11:48 +0200 (CEST) X-Ids: 164 Received: from [192.168.0.3] (mna75-2-81-57-226-176.fbx.proxad.net [81.57.226.176]) by smtp3-g19.free.fr (Postfix) with ESMTP id 25EFD487B9; Tue, 11 Apr 2006 10:11:47 +0200 (CEST) From: Annick Valibouze Organization: =?iso-8859-1?q?Universit=E9_Pierre_et_Marie?= Curie, Paris To: Jorge Barros de Abreu , annick.valibouze@upmc.fr Subject: Re: symmetries.texi Date: Tue, 11 Apr 2006 10:10:40 +0200 User-Agent: KMail/1.8 References: <200604091659.44544.ficmatin01@solar.com.br> In-Reply-To: <200604091659.44544.ficmatin01@solar.com.br> MIME-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Message-Id: <200604111010.40402.annick.valibouze@upmc.fr> X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-2.0.2 (shiva.jussieu.fr [134.157.0.164]); Tue, 11 Apr 2006 10:11:48 +0200 (CEST) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Miltered: at shiva.jussieu.fr with ID 443B64C4.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)! Status: R X-Status: NC X-KMail-EncryptionState: X-KMail-SignatureState: X-KMail-MDN-Sent: Hi, You will find the explanation in bottom SYM has a more recent version with more functions. I will transmit it to sourceforce when I will have the time with a copie for you. Thanks to transmit to me your last translation when you will have finish. I will give your site in my WEB page too. Whilliam Schelter, Bill, was one of my best friends.=20 2 or 4 months by year he lived on my house in Paris.=20 I am very touched that Maxima survive to him. I am very touched that Maxima is translated in local langage and is in this manner an internationnal Computer Algebra.=20 Bill was canadian and spoke french. I'm sure that I thinks (where he is) that this international translation of= =20 Maxima possible since Maxima is free (he has made all for that!) is excellent for the scientific knowledge. Thank you for your investissment. Sincerly yours, Annick Valibouze Professor Universit=C3=A9 Pierre et Marie Curie Paris France Page personnelle : http://www.calfor.lip6.fr/~avb (ou bien : http://www.lsta.upmc.fr) To answer, I fix the following Notation =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D= =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D Let f be an univariate polynomial on degree n over k with roots a_1,a_2,...a_n. Let P be a multivariate polynomial on m <=3D n variables x_1,...,x_m over k The symmetric group S_n of degree n acts naturally on P. Let S_n.P be the set of the permuted polynomials from P by S_n ex. S_3.(x_1 + x_2)=3D{x_1+x_2,x_1+x_3,x_2+x_3} The (absolute) resolvent of f by P is the univariate polynomial R(f,P) =3D prod (x - Q(a_1,q_2,...,a_n)) where Q belongs to S_n.P. Note H the subgroup of S_n containing the permutations of S_n which leaves = P=20 invariant ; i.e. the stabilisator of P in S_n. Now I can answer : =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D > *********************************** > Si la fonction f est unitaire : > > * un polyno^me d'une variable, =20 m=3D1 * line'aire ,=20 deg_{x_i}(P) <=3D 1 > * alterne'e, =20 P is the vandermond prod(x_i-x_j) where 1<=3D i < j <=3Dm > * une somme de variables,=20 x_1+\cdots + x_m > * syme'trique en les variables qui apparaissent dans son expression, symmetric polynomial on m variables > * un produit de variables, x1 ... x_m > * la fonction de la re'solvante de Cayley (utilisable qu'en degre' 5) > n=3Dm=3D5 =20 > and P=3D (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 - (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2 This last resolvent is without multiple roots if f it is and in this case if the Cayley resolvent has a simple root then f is solvable by radicals. Le Dimanche 9 Avril 2006 21:59, vous avez =C3=A9crit=C2=A0: > Hi Mr. Annick Valibouze > > I am a maxima translator to portuguese and translate your maxima manual > file "Symmetries.texi". The translation is in attachment. > > During the translation i have doubt about this part: > *********************************** > Si la fonction f est unitaire : > > =C2=A0 =C2=A0 * un polyno^me d'une variable, > =C2=A0 =C2=A0 * line'aire , > =C2=A0 =C2=A0 * alterne'e, > =C2=A0 =C2=A0 * une somme de variables, > =C2=A0 =C2=A0 * syme'trique en les variables qui apparaissent dans son ex= pression, > =C2=A0 =C2=A0 * un produit de variables, > =C2=A0 =C2=A0 * la fonction de la re'solvante de Cayley (utilisable qu'en= degre' 5) > =C2=A0 =C2=A0 =C2=A0 =C2=A0 > > (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 - > =C2=A0 =C2=A0 =C2=A0(x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2 > > =C2=A0 =C2=A0 =C2=A0 generale, > > le drapeau de resolvante pourra e^tre respectivement : > > =C2=A0 =C2=A0 * unitaire, > =C2=A0 =C2=A0 * lineaire, > =C2=A0 =C2=A0 * alternee, > =C2=A0 =C2=A0 * somme, > =C2=A0 =C2=A0 * produit, > =C2=A0 =C2=A0 * cayley, > =C2=A0 =C2=A0 * generale. > ********************************** > > > The correspondence is not clear for me: > > =C2=A0 =C2=A0 1 un polyno^me d'une variable, > =C2=A0 =C2=A0 2 line'aire , > =C2=A0 =C2=A0 3 alterne'e, > =C2=A0 =C2=A0 4 une somme de variables, > =C2=A0 =C2=A0 * syme'trique en les variables qui apparaissent dans son ex= pression, > =C2=A0 =C2=A0 5 un produit de variables, > =C2=A0 =C2=A0 6 la fonction de la re'solvante de Cayley (utilisable qu'en= degre' 5) > =C2=A0 =C2=A0 =C2=A0 =C2=A0 > > > =C2=A0 =C2=A0 =C2=A0 generale, > > =C2=A0 =C2=A0 1 unitaire, > =C2=A0 =C2=A0 2 lineaire, > =C2=A0 =C2=A0 3 alternee, > =C2=A0 =C2=A0 4 somme, > =C2=A0 =C2=A0 5 produit, > =C2=A0 =C2=A0 6 cayley, > =C2=A0 =C2=A0 * generale. > > Can you help me for clear this correspondence? > Isn't There correspondence for * ? > > Thanks in advance for your time. =2D-=20 Annick Valibouze =2D----------- Universit=C3=A9 Pierre et Marie Curie Paris France Page personnelle : http://www.calfor.lip6.fr/~avb (ou bien : http://www.lsta.upmc.fr) From ficmatin01@solar.com.br Sun Apr 9 21:59:44 2006 Return-Path: Delivered-To: online.fr-avbumpc@free.fr Received: (qmail 17995 invoked from network); 9 Apr 2006 20:01:07 -0000 Received: from shiva.jussieu.fr (134.157.0.129) by mrelay3-1.free.fr with SMTP; 9 Apr 2006 20:01:07 -0000 Received: from smtp3.pop.com.br (smtp3.pop.com.br [200.175.8.37]) by shiva.jussieu.fr (8.13.6/jtpda-5.4) with ESMTP id k39K13ws097317 for ; Sun, 9 Apr 2006 22:01:04 +0200 (CEST) X-Ids: 165 Received: from localhost (localhost [127.0.0.1]) by relay1-10025 (Postfix) with ESMTP id B082D5C1F5 for ; Sun, 9 Apr 2006 17:00:43 -0300 (EST) Received: from smtp3.pop.com.br (localhost [127.0.0.1]) by smtp3 (WCVirscan) with SMTP id 00001d4d443967eb ; Sun, 09 Apr 2006 17:00:43 -0300 Received: from filho1.server.net (200.175.188.50.dialup.gvt.net.br [200.175.188.50]) by relay1-25 (Postfix) with ESMTP id B9C4E8456 for ; Sun, 9 Apr 2006 17:00:22 -0300 (EST) From: Jorge Barros de Abreu Organization: Professor do Ensino =?iso-8859-1?q?M=E9dio?= To: annick.valibouze@upmc.fr Subject: symmetries.texi Date: Sun, 9 Apr 2006 16:59:44 -0300 User-Agent: KMail/1.7.2 MIME-Version: 1.0 Content-Disposition: inline Message-Id: <200604091659.44544.ficmatin01@solar.com.br> Content-Type: Multipart/Mixed; boundary="Boundary-00=_weWOE8O/YVSKTdi" X-DCC-POPInternet-Metrics: dcc 1259; Body=1 Fuz1=1 Fuz2=1 X-Greylist: IP, sender and recipient auto-whitelisted, not delayed by milter-greylist-1.7.2 (shiva.jussieu.fr [134.157.0.165]); Sun, 09 Apr 2006 22:01:05 +0200 (CEST) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Miltered: at shiva.jussieu.fr with ID 443967FF.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)! Status: R X-Status: NT X-KMail-EncryptionState: X-KMail-SignatureState: X-KMail-MDN-Sent: --Boundary-00=_weWOE8O/YVSKTdi Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Hi Mr. Annick Valibouze I am a maxima translator to portuguese and translate your maxima manual fil= e=20 "Symmetries.texi". The translation is in attachment. During the translation i have doubt about this part: *********************************** Si la fonction f est unitaire : =A0 =A0 * un polyno^me d'une variable, =A0 =A0 * line'aire , =A0 =A0 * alterne'e, =A0 =A0 * une somme de variables, =A0 =A0 * syme'trique en les variables qui apparaissent dans son expression, =A0 =A0 * un produit de variables, =A0 =A0 * la fonction de la re'solvante de Cayley (utilisable qu'en degre' = 5) =A0 =A0 =A0 =A0=20 (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 - =A0 =A0 =A0(x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2 =A0 =A0 =A0 generale,=20 le drapeau de resolvante pourra e^tre respectivement : =A0 =A0 * unitaire, =A0 =A0 * lineaire, =A0 =A0 * alternee, =A0 =A0 * somme, =A0 =A0 * produit, =A0 =A0 * cayley, =A0 =A0 * generale.=20 ********************************** The correspondence is not clear for me: =A0 =A0 1 un polyno^me d'une variable, =A0 =A0 2 line'aire , =A0 =A0 3 alterne'e, =A0 =A0 4 une somme de variables, =A0 =A0 * syme'trique en les variables qui apparaissent dans son expression, =A0 =A0 5 un produit de variables, =A0 =A0 6 la fonction de la re'solvante de Cayley (utilisable qu'en degre' = 5) =A0 =A0 =A0 =A0=20 =A0 =A0 =A0 generale,=20 =A0 =A0 1 unitaire, =A0 =A0 2 lineaire, =A0 =A0 3 alternee, =A0 =A0 4 somme, =A0 =A0 5 produit, =A0 =A0 6 cayley, =A0 =A0 * generale.=20 Can you help me for clear this correspondence? Isn't There correspondence for * ? Thanks in advance for your time. =2D-=20 Data Estelar 2453835.315532 http://www.solar.com.br/~ficmatin Desejo-lhe Paz, Vida Longa e Prosperidade. S=E3o Bem Vindas Mensagens no Formato Texto Gen=E9rico com Acentos. --Boundary-00=_weWOE8O/YVSKTdi Content-Type: text/plain; charset="iso-8859-1"; name="Symmetries.texi" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="Symmetries.texi" @c Language: Portuguese, Encoding: iso-8859-1 @c /Symmetries.texi/1.9/Thu Apr 21 04:58:35 2005/-ko/ @c arquivo gentilmente traduzido por Helciclever Barros da Silva @c end concepts Symmetries @menu * Defini@value{cedilha}@~oes para Simetrias:: @end menu @node Defini@value{cedilha}@~oes para Simetrias, , Simetrias, Simetrias @section Defini@value{cedilha}@~oes para Simetrias @deffn {Fun@,{c}@~ao} comp2pui (@var{n}, @var{l}) realiza a passagem das fun@,{c}@~oes sim@'etricas completas, dadas na lista @var{l}, @`as fun@,{c}@~oes sim@'etricas elementares de 0 a @var{n}. Se a lista @var{l} cont@'em menos de @code{@var{n}+1} elementos os valores formais v@^em complet@'a-los. O primeiro elemento da lista @var{l} fornece o cardinal do alfabeto se ele existir, se n@~ao existir coloca-se igual a @var{n}. @c GENERATED FROM THE FOLLOWING @c comp2pui (3, [4, g]); @example (%i1) comp2pui (3, [4, g]); 2 2 (%o1) [4, g, 2 h2 - g , 3 h3 - g h2 + g (g - 2 h2)] @end example @end deffn @deffn {Fun@,{c}@~ao} cont2part (@var{pc}, @var{lvar}) Torna o polin@^omio particionado associado @`a forma contra@'ida @var{pc} cujas vari@'aveis est@~ao em @var{lvar}. @c GENERATED FROM THE FOLLOWING @c pc: 2*a^3*b*x^4*y + x^5; @c cont2part (pc, [x, y]); @example (%i1) pc: 2*a^3*b*x^4*y + x^5; 3 4 5 (%o1) 2 a b x y + x (%i2) cont2part (pc, [x, y]); 3 (%o2) [[1, 5, 0], [2 a b, 4, 1]] @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}a de representa@,{c}@~ao : @code{contract}, @code{explose}, @code{part2cont}, @code{partpol}, @code{tcontract}, @code{tpartpol}. @end deffn @deffn {Fun@,{c}@~ao} contract (@var{psym}, @var{lvar}) torna uma forma contra@'ida (i.e. um mon@^omio por @'orbita sobre a a@,{c}@~ao do grupo sim@'etrico) do polin@^omio @var{psym} em vari@'aveis contidas na lista @var{lvar}. A fun@,{c}@~ao @code{explose} realisa a opera@,{c}@~ao inversa. A fun@,{c}@~ao @code{tcontract} testa adicionalmente a simetria do polin@^omio. @c GENERATED FROM THE FOLLOWING @c psym: explose (2*a^3*b*x^4*y, [x, y, z]); @c contract (psym, [x, y, z]); @example (%i1) psym: explose (2*a^3*b*x^4*y, [x, y, z]); 3 4 3 4 3 4 3 4 (%o1) 2 a b y z + 2 a b x z + 2 a b y z + 2 a b x z 3 4 3 4 + 2 a b x y + 2 a b x y (%i2) contract (psym, [x, y, z]); 3 4 (%o2) 2 a b x y @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}a de representa@,{c}@~ao : @code{cont2part}, @code{explose}, @code{part2cont}, @code{partpol}, @code{tcontract}, @code{tpartpol}. @end deffn @deffn {Fun@,{c}@~ao} direct ([@var{p_1}, ..., @var{p_n}], @var{y}, @var{f}, [@var{lvar_1}, ..., @var{lvar_n}]) calcula a im@'agem direta (veja M. GIUSTI, D. LAZARD et A. VALIBOUZE, ISSAC 1988, Rome) associada @`a fun@,{c}@~ao @var{f}, nas listas de vari@'aveis @var{lvar_1}, ..., @var{lvar_n}, e nos polin@^omios @var{p_1}, ..., @var{p_n} de uma vari@'avel @var{y}. l'arite' da fun@,{c}@~ao @var{f} @'e importante para o c@'alculo. Assim, se a express@~ao de @var{f} n@~ao depende de uma vari@'avel, n@~ao somente @'e in@'util fornecer essa vari@'avel como tamb@'em diminui consideravelmente os c@'alculos se a vari@'avel n@~ao for fornecida. @c GENERATED FROM THE FOLLOWING @c direct ([z^2 - e1* z + e2, z^2 - f1* z + f2], @c z, b*v + a*u, [[u, v], [a, b]]); @c ratsimp (%); @c ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2 - f1* z + f2], @c z, b*v + a*u, [[u, v], [a, b]])); @example (%i1) direct ([z^2 - e1* z + e2, z^2 - f1* z + f2], z, b*v + a*u, [[u, v], [a, b]]); 2 (%o1) y - e1 f1 y 2 2 2 2 - 4 e2 f2 - (e1 - 2 e2) (f1 - 2 f2) + e1 f1 + ----------------------------------------------- 2 (%i2) ratsimp (%); 2 2 2 (%o2) y - e1 f1 y + (e1 - 4 e2) f2 + e2 f1 (%i3) ratsimp (direct ([z^3-e1*z^2+e2*z-e3,z^2 - f1* z + f2], z, b*v + a*u, [[u, v], [a, b]])); 6 5 2 2 2 4 (%o3) y - 2 e1 f1 y + ((2 e1 - 6 e2) f2 + (2 e2 + e1 ) f1 ) y 3 3 3 + ((9 e3 + 5 e1 e2 - 2 e1 ) f1 f2 + (- 2 e3 - 2 e1 e2) f1 ) y 2 2 4 2 + ((9 e2 - 6 e1 e2 + e1 ) f2 2 2 2 2 4 + (- 9 e1 e3 - 6 e2 + 3 e1 e2) f1 f2 + (2 e1 e3 + e2 ) f1 ) 2 2 2 3 2 y + (((9 e1 - 27 e2) e3 + 3 e1 e2 - e1 e2) f1 f2 2 2 3 5 + ((15 e2 - 2 e1 ) e3 - e1 e2 ) f1 f2 - 2 e2 e3 f1 ) y 2 3 3 2 2 3 + (- 27 e3 + (18 e1 e2 - 4 e1 ) e3 - 4 e2 + e1 e2 ) f2 2 3 3 2 2 + (27 e3 + (e1 - 9 e1 e2) e3 + e2 ) f1 f2 2 4 2 6 + (e1 e2 e3 - 9 e3 ) f1 f2 + e3 f1 @end example Pesquisa de polin@^omios cujas ra@'izes s@~ao a soma a+u ou a @'e a ra@'iz de z^2 - e1* z + e2 e u @'e a ra@'iz de z^2 - f1* z + f2 @c GENERATED FROM THE FOLLOWING @c ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2], @c z, a + u, [[u], [a]])); @example (%i1) ratsimp (direct ([z^2 - e1* z + e2, z^2 - f1* z + f2], z, a + u, [[u], [a]])); 4 3 2 (%o1) y + (- 2 f1 - 2 e1) y + (2 f2 + f1 + 3 e1 f1 + 2 e2 2 2 2 2 + e1 ) y + ((- 2 f1 - 2 e1) f2 - e1 f1 + (- 2 e2 - e1 ) f1 2 2 2 - 2 e1 e2) y + f2 + (e1 f1 - 2 e2 + e1 ) f2 + e2 f1 + e1 e2 f1 2 + e2 @end example @code{direct} pode assumir dois sinalizadores: @code{elementaires} (elementares) e @code{puissances} (exponenciais - valor padr@~ao) que permitem a decomposi@,{c}@~ao de polin@^omios sim@'etricos que aparecerem nesses c@'alculos pelas fun@,{c}@~oes sim@'etricas elementares ou pelas fun@,{c}@~oes exponenciais respectivamente. Fun@,{c}@~oes de @code{sym} utilizadas nesta fun@,{c}@~ao : @code{multi_orbit} (portanto @code{orbit}), @code{pui_direct}, @code{multi_elem} (portanto @code{elem}), @code{multi_pui} (portanto @code{pui}), @code{pui2ele}, @code{ele2pui} (se o sinalizador @code{direct} for escolhido para @code{puissances}). @end deffn @deffn {Fun@,{c}@~ao} ele2comp (@var{m}, @var{l}) passa das fun@,{c}@~oes sim@'etricas elementares para fun@,{c}@~oes completas. Semelhante a @code{comp2ele} e a @code{comp2pui}. Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2pui}, @code{elem}, @code{mon2schur}, @code{multi_elem}, @code{multi_pui}, @code{pui}, @code{pui2comp}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} ele2polynome (@var{l}, @var{z}) fornece o polin@^omio em @var{z} cujas fun@,{c}@~oes sim@'etricas elementares das ra@'izes estiverem na lista @var{l}. @code{@var{l} = [@var{n}, @var{e_1}, ..., @var{e_n}]} onde @var{n} @'e o grau do polin@^omio e @var{e_i} @'e a @var{i}-@'esima fun@,{c}@~ao sim@'etrica elementar. @c GENERATED FROM THE FOLLOWING @c ele2polynome ([2, e1, e2], z); @c polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x); @c ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x); @example (%i1) ele2polynome ([2, e1, e2], z); 2 (%o1) z - e1 z + e2 (%i2) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x); (%o2) [7, 0, - 14, 0, 56, 0, - 56, - 22] (%i3) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x); 7 5 3 (%o3) x - 14 x + 56 x - 56 x + 22 @end example @noindent A rec@'iproca: @code{polynome2ele (@var{P}, @var{z})} Veja tamb@'em: @code{polynome2ele}, @code{pui2polynome}. @end deffn @deffn {Fun@,{c}@~ao} ele2pui (@var{m}, @var{l}) passa de fun@,{c}@~oes sim@'etricas elementares para fun@,{c}@~oes completas. Similar a @code{comp2ele} e @code{comp2pui}. Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{elem}, @code{mon2schur}, @code{multi_elem}, @code{multi_pui}, @code{pui}, @code{pui2comp}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} elem (@var{ele}, @var{sym}, @var{lvar}) decomp@~oe o polin@^omio sim@'etrico @var{sym}, nas vari@'aveis cont@'inuas da lista @var{lvar}, em fun@,{c}@~oes sim@'etricas elementares contidas na lista @var{ele}. Se o primeiro elemento de @var{ele} for fornecido esse ser@'a o cardinal do alfabeto se n@~ao for utilizado o grau do polin@^omio @var{sym}. Se falta valores para a lista @var{ele} valores formais do tipo "ei" s@~ao novamente colocados para completar a lista. O polin@^omio @var{sym} pode ser fornecido de 3 formas diferentes : contra@'ida (@code{elem} deve protanto valer 1 que @'e seu valor padr@~ao), particionada (@code{elem} deve valer 3) ou extendida (i.e. o polin@^omio por completo) (@code{elem} deve valer 2). A utiliza@,{c}@~ao da fun@,{c}@~ao @code{pui} se realiza sobre o mesmo modelo. Sob um alfabeto de cardinal 3 com @var{e1}, a primeira fun@,{c}@~ao sim@'etrica elementar, valendo 7, o polin@^omio sim@'etrico em 3 vari@'aveis cuja forma contra@'ida (aqui, s@'o depende de duas de suas vari@'aveis) @'e x^4-2*x*y decomp@~oe-se em fun@,{c}@~oes sim@'etricas elementares : @c GENERATED FROM THE FOLLOWING @c elem ([3, 7], x^4 - 2*x*y, [x, y]); @c ratsimp (%); @example (%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]); (%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3 + (- 2 (49 - e2) - 2) e2 (%i2) ratsimp (%); 2 (%o2) 28 e3 + 2 e2 - 198 e2 + 2401 @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{ele2pui}, @code{mon2schur}, @code{multi_elem}, @code{multi_pui}, @code{pui}, @code{pui2comp}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} explose (@var{pc}, @var{lvar}) toma o polin@^omio sim@'etrico associado @`a forma contra@'ida @var{pc}. A lista @var{lvar} cont@'em vari@'aveis. @c GENERATED FROM THE FOLLOWING @c explose (a*x + 1, [x, y, z]); @example (%i1) explose (a*x + 1, [x, y, z]); (%o1) a z + a y + a x + 1 @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}a de representa@,{c}@~ao : @code{contract}, @code{cont2part}, @code{part2cont}, @code{partpol}, @code{tcontract}, @code{tpartpol}. @end deffn @deffn {Fun@,{c}@~ao} kostka (@var{part_1}, @var{part_2}) escrita por P. ESPERET, calcula o n@'umero de Kostka associado @`as parti@,{c}@~oes @var{part_1} e @var{part_2}. @c GENERATED FROM THE FOLLOWING @c kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]); @example (%i1) kostka ([3, 3, 3], [2, 2, 2, 1, 1, 1]); (%o1) 6 @end example @end deffn @deffn {Fun@,{c}@~ao} lgtreillis (@var{n}, @var{m}) torna a lista de parti@,{c}@~oes de peso @var{n} e de largura @var{m}. @c GENERATED FROM THE FOLLOWING @c lgtreillis (4, 2); @example (%i1) lgtreillis (4, 2); (%o1) [[3, 1], [2, 2]] @end example Veja tamb@'em : @code{ltreillis}, @code{treillis} e @code{treinat}. @end deffn @deffn {Fun@,{c}@~ao} ltreillis (@var{n}, @var{m}) torna a lista de parti@,{c}@~oes de peso @var{n} e largura menor ou igual a @var{m}. @c GENERATED FROM THE FOLLOWING @c ltreillis (4, 2); @example (%i1) ltreillis (4, 2); (%o1) [[4, 0], [3, 1], [2, 2]] @end example @noindent Veja tamb@'em : @code{lgtreillis}, @code{treillis} e @code{treinat}. @end deffn @c NOT REALLY HAPPY ABOUT MATH NOTATION HERE @deffn {Fun@,{c}@~ao} mon2schur (@var{l}) A lista @var{l} representa a fun@,{c}@~ao de Schur S_@var{l}: @c On a = sendo Temos @var{l} = [@var{i_1}, @var{i_2}, ..., @var{i_q}] com @var{i_1} <= @var{i_2} <= ... <= @var{i_q}. A fun@,{c}@~ao de Schur @'e S_[@var{i_1}, @var{i_2}, ..., @var{i_q}] @'e a menor da mariz infinita (h_@{i-j@}) @var{i} >= 1, @var{j} >= 1 composta das @var{q} primeiras linhas e de colunas @var{i_1} + 1, @var{i_2} + 2, ..., @var{i_q} + @var{q}. Escreve-se essa fun@,{c}@~ao de Schur em fun@,{c}@~ao das formas monomiais utilizando as fun@,{c}@~oes @code{treinat} e @code{kostka}. A forma retornada @'e um polin@^omio sim@'etrico em uma de suas representa@,{c}@~oes contra@'idas com as vari@'aveis @var{x_1}, @var{x_2}, .... @c GENERATED FROM THE FOLLOWING @c mon2schur ([1, 1, 1]); @c mon2schur ([3]); @c mon2schur ([1, 2]); @example (%i1) mon2schur ([1, 1, 1]); (%o1) x1 x2 x3 (%i2) mon2schur ([3]); 2 3 (%o2) x1 x2 x3 + x1 x2 + x1 (%i3) mon2schur ([1, 2]); 2 (%o3) 2 x1 x2 x3 + x1 x2 @end example @noindent queremos dizer que para 3 vari@'aveis tem-se : @c UM, FROM WHAT ARGUMENTS WAS THE FOLLOWING GENERATED ?? @example 2 x1 x2 x3 + x1^2 x2 + x2^2 x1 + x1^2 x3 + x3^2 x1 + x2^2 x3 + x3^2 x2 @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{ele2pui}, @code{elem}, @code{multi_elem}, @code{multi_pui}, @code{pui}, @code{pui2comp}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} multi_elem (@var{l_elem}, @var{multi_pc}, @var{l_var}) decomp@~oe um polin@^omio multi-sim@'etrico sob a forma multi-contra@'ida @var{multi_pc} nos grupos de vari@'aveis contidas na lista de listas @var{l_var} sobre os groupos de fun@,{c}@~oes sim@'etricas elementares contidas em @var{l_elem}. @c GENERATED FROM THE FOLLOWING @c multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3, [[x, y], [a, b]]); @c ratsimp (%); @example (%i1) multi_elem ([[2, e1, e2], [2, f1, f2]], a*x + a^2 + x^3, [[x, y], [a, b]]); 3 (%o1) - 2 f2 + f1 (f1 + e1) - 3 e1 e2 + e1 (%i2) ratsimp (%); 2 3 (%o2) - 2 f2 + f1 + e1 f1 - 3 e1 e2 + e1 @end example Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{ele2pui}, @code{elem}, @code{mon2schur}, @code{multi_pui}, @code{pui}, @code{pui2comp}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} multi_orbit (@var{P}, [@var{lvar_1}, @var{lvar_2}, ..., @var{lvar_p}]) @var{P} @'e um polin@^omio no conjunto das vari@'aveis contidas nas listas @var{lvar_1}, @var{lvar_2}, ..., @var{lvar_p}. Essa fun@,{c}@~ao leva novamente na @'orbita do polin@^omio @var{P} sob a a@,{c}@~ao do do produto dos grupos sim@'etricos dos conjuntos de vari@'aveis representados por essas @var{p} listas. @c GENERATED FROM THE FOLLOWING @c multi_orbit (a*x + b*y, [[x, y], [a, b]]); @c multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]); @example (%i1) multi_orbit (a*x + b*y, [[x, y], [a, b]]); (%o1) [b y + a x, a y + b x] (%i2) multi_orbit (x + y + 2*a, [[x, y], [a, b, c]]); (%o2) [y + x + 2 c, y + x + 2 b, y + x + 2 a] @end example @noindent Veja tamb@'em : @code{orbit} pela a@,{c}@~ao de um s@'o grupo sim@'etrico. @end deffn @c WHAT ARE THE ARGUMENTS FOR THIS FUNCTION ?? @deffn {Fun@,{c}@~ao} multi_pui est@'a para a fun@,{c}@~ao @code{pui} da mesma forma que a fun@,{c}@~ao @code{multi_elem} est@'a para a fun@,{c}@~ao @code{elem}. @c GENERATED FROM THE FOLLOWING @c multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3, [[x, y], [a, b]]); @example (%i1) multi_pui ([[2, p1, p2], [2, t1, t2]], a*x + a^2 + x^3, [[x, y], [a, b]]); 3 3 p1 p2 p1 (%o1) t2 + p1 t1 + ------- - --- 2 2 @end example @end deffn @deffn {Fun@,{c}@~ao} multinomial (@var{r}, @var{part}) onde @var{r} @'e o peso da parti@,{c}@~ao @var{part}. Essa fun@,{c}@~ao reporta ao coeficiente multinomial associado : se as partes das parti@,{c}@~oes @var{part} forem @var{i_1}, @var{i_2}, ..., @var{i_k}, o resultado de @code{multinomial} @'e @code{@var{r}!/(@var{i_1}! @var{i_2}! ... @var{i_k}!)}. @end deffn @deffn {Fun@,{c}@~ao} multsym (@var{ppart_1}, @var{ppart_2}, @var{n}) realiza o produto de dois polin@^omios sim@'etricos de @var{n} vari@'aveis s@'o trabalhando o m@'odulo da a@,{c}@~ao do grupo sim@'etrico de ordem @var{n}. Os polin@^omios est@~ao em sua representa@,{c}@~ao particionada. Sejam os 2 polin@^omios sim@'etricos em @code{x}, @code{y}: @code{3*(x + y) + 2*x*y} e @code{5*(x^2 + y^2)} cujas formas particionada s@~ao respectivamente @code{[[3, 1], [2, 1, 1]]} e @code{[[5, 2]]}, ent@~ao seu produto ser@'a dado por : @c GENERATED FROM THE FOLLOWING @c multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2); @example (%i1) multsym ([[3, 1], [2, 1, 1]], [[5, 2]], 2); (%o1) [[10, 3, 1], [15, 3, 0], [15, 2, 1]] @end example @noindent seja @code{10*(x^3*y + y^3*x) + 15*(x^2*y + y^2*x) + 15*(x^3 + y^3)}. Fun@,{c}@~oes de mudan@,{c}a de representa@,{c}@~ao de um polin@^omio sim@'etrico : @code{contract}, @code{cont2part}, @code{explose}, @code{part2cont}, @code{partpol}, @code{tcontract}, @code{tpartpol}. @end deffn @deffn {Fun@,{c}@~ao} orbit (@var{P}, @var{lvar}) calcula a @'orbita de um polin@^omio @var{P} nas vari@'aveis da lista @var{lvar} soba a a@,{c}@~ao do grupo sim@'etrico do conjunto das vari@'aveis contidas na lista @var{lvar}. @c GENERATED FROM THE FOLLOWING @c orbit (a*x + b*y, [x, y]); @c orbit (2*x + x^2, [x, y]); @example (%i1) orbit (a*x + b*y, [x, y]); (%o1) [a y + b x, b y + a x] (%i2) orbit (2*x + x^2, [x, y]); 2 2 (%o2) [y + 2 y, x + 2 x] @end example @noindent Veja tamb@'em : @code{multi_orbit} para a a@,{c}@~ao de um produto de grupos sim@'etricos sobre um polin@^omio. @end deffn @deffn {Fun@,{c}@~ao} part2cont (@var{ppart}, @var{lvar}) passa da form particionada @`a forma contra@'ida d um polin@^omio sim@'etrico. A forma contra@'ida @'e conseguida com as vari@'aveis contidas em @var{lvar}. @c GENERATED FROM THE FOLLOWING @c part2cont ([[2*a^3*b, 4, 1]], [x, y]); @example (%i1) part2cont ([[2*a^3*b, 4, 1]], [x, y]); 3 4 (%o1) 2 a b x y @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}a de representa@,{c}@~ao : @code{contract}, @code{cont2part}, @code{explose}, @code{partpol}, @code{tcontract}, @code{tpartpol}. @end deffn @deffn {Fun@,{c}@~ao} partpol (@var{psym}, @var{lvar}) @var{psym} @'e um polin@^omio sim@'etrico nas vari@'aveis de @var{lvar}. Esta fun@,{c}@~ao retoma sua representa@,{c}@~ao particionada. @c GENERATED FROM THE FOLLOWING @c partpol (-a*(x + y) + 3*x*y, [x, y]); @example (%i1) partpol (-a*(x + y) + 3*x*y, [x, y]); (%o1) [[3, 1, 1], [- a, 1, 0]] @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}a de representa@,{c}@~ao : @code{contract}, @code{cont2part}, @code{explose}, @code{part2cont}, @code{tcontract}, @code{tpartpol}. @end deffn @deffn {Fun@,{c}@~ao} permut (@var{l}) retoma a lista de permuta@,{c}@~oes da lista @var{l}. @end deffn @deffn {Fun@,{c}@~ao} polynome2ele (@var{P}, @var{x}) fornece a lista @code{@var{l} = [@var{n}, @var{e_1}, ..., @var{e_n}]} onde @var{n} @'e o grau do polin@^omio @var{P} na vari@'avel @var{x} e @var{e_i} @'e a @var{i}-@'ezima fun@,{c}@~ao sim@'etrica elementar das ra@'izes de @var{P}. @c GENERATED FROM THE FOLLOWING @c polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x); @c ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x); @example (%i1) polynome2ele (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x); (%o1) [7, 0, - 14, 0, 56, 0, - 56, - 22] (%i2) ele2polynome ([7, 0, -14, 0, 56, 0, -56, -22], x); 7 5 3 (%o2) x - 14 x + 56 x - 56 x + 22 @end example @noindent A rec@'iproca : @code{ele2polynome (@var{l}, @var{x})} @end deffn @deffn {Fun@,{c}@~ao} prodrac (@var{l}, @var{k}) @var{l} @'e uma lista que cont@'em as fun@,{c}@~oes sim@'etricas elementares sob um conjunto @var{A}. @code{prodrac} produz o polin@^omio cujas ra@'izes s@~ao os produtos @var{k} a @var{k} dos elementos de @var{A}. @end deffn @c HMM, pui IS A VARIABLE AS WELL @deffn {Fun@,{c}@~ao} pui (@var{l}, @var{sym}, @var{lvar}) decomp@~oe o polin@^omio sim@'etrico @var{sym}, nas vari@'aveis contidas a lista @var{lvar}, nas fun@,{c}@~oes exponenciais contidas na lista @var{l}. Se o primeiro elemento de @var{l} for dado ele ser@'a o cardinal do alfabeto se n@~ao for dado toma-se o grau do polin@^omio @var{sym} para ser o cardinal do alfabeto. Se faltarem valores na lista @var{l}, valores formais do typo "pi" ser@~ao colocados na lista. O polin@^omio @code{sym} pode ser dado sob 3 formas diferentes : contra@'ida (@code{pui} deve valer 1 - seu valor padr@~ao), particionada (@code{pui} deve valer 3) ou estendida (i.e. o polin@^omio por completo) (@code{pui} deve valer 2). A fun@,{c}@~ao @code{elem} se utiliza da mesma maneira. @c GENERATED FROM THE FOLLOWING @c pui; @c pui ([3, a, b], u*x*y*z, [x, y, z]); @c ratsimp (%); @example (%i1) pui; (%o1) 1 (%i2) pui ([3, a, b], u*x*y*z, [x, y, z]); 2 a (a - b) u (a b - p3) u (%o2) ------------ - ------------ 6 3 (%i3) ratsimp (%); 3 (2 p3 - 3 a b + a ) u (%o3) --------------------- 6 @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{ele2pui}, @code{elem}, @code{mon2schur}, @code{multi_elem}, @code{multi_pui}, @code{pui2comp}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} pui2comp (@var{n}, @var{lpui}) produz a lista das @var{n} primeiras fun@,{c}@~oes completas (com o cardinal em primeiro lugar) em fun@,{c}@~ao das fun@,{c}@~oes exponenciais dadas na lista @var{lpui}. Se a lista @var{lpui} estiver vazia o cardianl ser@'a N, se n@~ao estiver vazia, ser@'a o primeiro elemento de forma an@'aloga a @code{comp2ele} e a @code{comp2pui}. @c GENERATED FROM THE FOLLOWING @c pui2comp (2, []); @c pui2comp (3, [2, a1]); @c ratsimp (%); @example (%i1) pui2comp (2, []); 2 p2 + p1 (%o1) [2, p1, --------] 2 (%i2) pui2comp (3, [2, a1]); 2 a1 (p2 + a1 ) 2 p3 + ------------- + a1 p2 p2 + a1 2 (%o2) [2, a1, --------, --------------------------] 2 3 (%i3) ratsimp (%); 2 3 p2 + a1 2 p3 + 3 a1 p2 + a1 (%o3) [2, a1, --------, --------------------] 2 6 @end example @noindent Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{ele2pui}, @code{elem}, @code{mon2schur}, @code{multi_elem}, @code{multi_pui}, @code{pui}, @code{pui2ele}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} pui2ele (@var{n}, @var{lpui}) realiza a transforma@,{c}@~ao das fun@,{c}@~oes exponenciais em fun@,{c}@~oes sim@'etricos elementares. Se o sinalizador @code{pui2ele} for @code{girard}, recupera-se a lista de fun@,{c}@~oes sim@'etricos elementares de 1 a @var{n}, e se for igual a @code{close}, recupera-se a @var{n}-@'ezima fun@,{c}@~ao sim@'etrica elementar. Outras fun@,{c}@~oes de mudan@,{c}as de base : @code{comp2ele}, @code{comp2pui}, @code{ele2comp}, @code{ele2pui}, @code{elem}, @code{mon2schur}, @code{multi_elem}, @code{multi_pui}, @code{pui}, @code{pui2comp}, @code{puireduc}, @code{schur2comp}. @end deffn @deffn {Fun@,{c}@~ao} pui2polynome (@var{x}, @var{lpui}) calcula o polin@^omio em @var{x} cujas fun@,{c}@~oes exponenciais das ra@'izes s@~ao dadas na lista @var{lpui}. @c GENERATED FROM THE FOLLOWING @c polynome2ele (x^3 - 4*x^2 + 5*x - 1, x); @c ele2pui (3, %); @c pui2polynome (x, %); @example (%i1) pui; (%o1) 1 (%i2) kill(labels); (%o0) done (%i1) polynome2ele (x^3 - 4*x^2 + 5*x - 1, x); (%o1) [3, 4, 5, 1] (%i2) ele2pui (3, %); (%o2) [3, 4, 6, 7] (%i3) pui2polynome (x, %); 3 2 (%o3) x - 4 x + 5 x - 1 @end example @noindent Autres fun@,{c}@~oes a` voir : @code{polynome2ele}, @code{ele2polynome}. @end deffn @deffn {Fun@,{c}@~ao} pui_direct (@var{orbite}, [@var{lvar_1}, ..., @var{lvar_n}], [@var{d_1}, @var{d_2}, ..., @var{d_n}]) Seja @var{f} um polin@^omio em @var{n} blocos de vari@'aveis @var{lvar_1}, ..., @var{lvar_n}. Seja @var{c_i} o n@'umero de vari@'aveis em @var{lvar_i} . E @var{SC} o produto dos @var{n} grupos sim@'etricos de grau @var{c_1}, ..., @var{c_n}. Esse grupo age naturalmente sobre @var{f}. A Lista @var{orbite} @'e a @'orbita, anotada de @code{@var{SC}(@var{f})}, da fun@,{c}@~ao @var{f} sob a a@,{c}@~ao de @var{SC}. (Essa lista pode ser obtida com a fun@,{c}@~ao : @code{multi_orbit}). Os @code{d_i} s@~ao inteiros tais que @var{c_1} <= @var{d_1}, @var{c_2} <= @var{d_2}, ..., @var{c_n} <= @var{d_n}. Seja @var{SD} o produto dos grupos sim@'etricos @var{S_d1} x @var{S_d2} x ... x @var{S_dn}. A fun@,{c}@~ao @code{pui_direct} retorna as @var{n} premeiras fun@,{c}@~oes exponenciais de @code{@var{SD}(@var{f})} dedzidas das fun@,{c}@~oes exponenciais de @code{@var{SC}(@var{f})} onde @var{n} @'e o cardinal de @code{@var{SD}(@var{f})}. O resultado @'e produzido sob a forma multi-contra@'ida em rela@,{c}@~ao a @var{SD}. i.e. apenas se conserva um elemento por @'orbita sob a a@,{c}@~ao de @var{SD}). @c GENERATED FROM THE FOLLOWING @c l: [[x, y], [a, b]]; @c pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]); @c pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]); @c pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a], [[x, y], [a, b, c]], [2, 3]); @example (%i1) l: [[x, y], [a, b]]; (%o1) [[x, y], [a, b]] (%i2) pui_direct (multi_orbit (a*x + b*y, l), l, [2, 2]); 2 2 (%o2) [a x, 4 a b x y + a x ] (%i3) pui_direct (multi_orbit (a*x + b*y, l), l, [3, 2]); 2 2 2 2 3 3 (%o3) [2 a x, 4 a b x y + 2 a x , 3 a b x y + 2 a x , 2 2 2 2 3 3 4 4 12 a b x y + 4 a b x y + 2 a x , 3 2 3 2 4 4 5 5 10 a b x y + 5 a b x y + 2 a x , 3 3 3 3 4 2 4 2 5 5 6 6 40 a b x y + 15 a b x y + 6 a b x y + 2 a x ] (%i4) pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a], [[x, y], [a, b, c]], [2, 3]); 2 2 (%o4) [3 x + 2 a, 6 x y + 3 x + 4 a x + 4 a , 2 3 2 2 3 9 x y + 12 a x y + 3 x + 6 a x + 12 a x + 8 a ] @end example @c THIS NEXT FUNCTION CALL TAKES A VERY LONG TIME (SEVERAL MINUTES) @c SO LEAVE IT OUT TIL PROCESSORS GET A LITTLE FASTER ... @c pui_direct ([y + x + 2*c, y + x + 2*b, y + x + 2*a], [[x, y], [a, b, c]], [3, 4]); @end deffn @deffn {Fun@,{c}@~ao} puireduc (@var{n}, @var{lpui}) @var{lpui} @'e uma lista cujo primeiro elemento @'e um inteiro @var{m}. @code{puireduc} fornece as @var{n} primeiras fun@,{c}@~oes exponenciais em fun@,{c}@~ao das @var{m} primeira. @c GENERATED FROM THE FOLLOWING @c puireduc (3, [2]); @example (%i1) puireduc (3, [2]); 2 p1 (p1 - p2) (%o1) [2, p1, p2, p1 p2 - -------------] 2 (%i2) ratsimp (%); 3 3 p1 p2 - p1 (%o2) [2, p1, p2, -------------] 2 @end example @end deffn @deffn {Fun@,{c}@~ao} resolvante (@var{P}, @var{x}, @var{f}, [@var{x_1}, ..., @var{x_d}]) calcula a resolvente do polin@^omio @var{P} em rela@,{c}@~ao @`a vari@'avel @var{x} e de grau @var{n} >= @var{d} pela fun@,{c}@~ao @var{f} expressa nas vari@'aveis @var{x_1}, ..., @var{x_d}. @'E importante para a efic@'acia dos c@'alculos n@~ao colocar na lista @code{[@var{x_1}, ..., @var{x_d}]} as vari@'aveis n@~ao interferindo na fun@,{c}@~ao de transforma@,{c}@~ao @var{f}. Afim de tornar mais eficazes os c@'alculos pode-se colocar sinalizadores na vari@'avel @code{resolvante} para que os algor@'itmos adequados sejam utilizados : Se a fun@,{c}@~ao @var{f} for unit@'aria : @itemize @bullet @item um polin@^omio de uma vari@'avel, @item linear , @item alternado, @item uma soma de vari@'aveis, @item sim@'etrico nas vari@'aveis que aparecem em sua express@~ao, @item um produto de vari@'aveis, @item a fun@,{c}@~ao da resolvente de Cayley (utilis@'avel no grau 5) @c WHAT IS THIS ILLUSTRATING EXACTLY ?? @example (x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x1 - (x1*x3 + x3*x5 + x5*x2 + x2*x4 + x4*x1))^2 @end example geral, @end itemize o sinalizador da @code{resolvante} poder@'a ser respectivamente : @itemize @bullet @item unitaire, @item lineaire, @item alternee, @item somme, @item produit, @item cayley, @item generale. @end itemize @c GENERATED FROM THE FOLLOWING @c resolvante: unitaire$ @c resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1, [x]); @c resolvante: lineaire$ @c resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]); @c resolvante: general$ @c resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]); @c resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]); @c direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]); @c resolvante :lineaire$ @c resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]); @c resolvante: symetrique$ @c resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]); @c resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]); @c resolvante: alternee$ @c resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]); @c resolvante: produit$ @c resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]); @c resolvante: symetrique$ @c resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]); @c resolvante: cayley$ @c resolvante (x^5 - 4*x^2 + x + 1, x, a, []); @example (%i1) resolvante: unitaire$ (%i2) resolvante (x^7 - 14*x^5 + 56*x^3 - 56*x + 22, x, x^3 - 1, [x]); " resolvante unitaire " [7, 0, 28, 0, 168, 0, 1120, - 154, 7840, - 2772, 56448, - 33880, 413952, - 352352, 3076668, - 3363360, 23114112, - 30494464, 175230832, - 267412992, 1338886528, - 2292126760] 3 6 3 9 6 3 [x - 1, x - 2 x + 1, x - 3 x + 3 x - 1, 12 9 6 3 15 12 9 6 3 x - 4 x + 6 x - 4 x + 1, x - 5 x + 10 x - 10 x + 5 x 18 15 12 9 6 3 - 1, x - 6 x + 15 x - 20 x + 15 x - 6 x + 1, 21 18 15 12 9 6 3 x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1] [- 7, 1127, - 6139, 431767, - 5472047, 201692519, - 3603982011] 7 6 5 4 3 2 (%o2) y + 7 y - 539 y - 1841 y + 51443 y + 315133 y + 376999 y + 125253 (%i3) resolvante: lineaire$ (%i4) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]); " resolvante lineaire " 24 20 16 12 8 (%o4) y + 80 y + 7520 y + 1107200 y + 49475840 y 4 + 344489984 y + 655360000 (%i5) resolvante: general$ (%i6) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3]); " resolvante generale " 24 20 16 12 8 (%o6) y + 80 y + 7520 y + 1107200 y + 49475840 y 4 + 344489984 y + 655360000 (%i7) resolvante (x^4 - 1, x, x1 + 2*x2 + 3*x3, [x1, x2, x3, x4]); " resolvante generale " 24 20 16 12 8 (%o7) y + 80 y + 7520 y + 1107200 y + 49475840 y 4 + 344489984 y + 655360000 (%i8) direct ([x^4 - 1], x, x1 + 2*x2 + 3*x3, [[x1, x2, x3]]); 24 20 16 12 8 (%o8) y + 80 y + 7520 y + 1107200 y + 49475840 y 4 + 344489984 y + 655360000 (%i9) resolvante :lineaire$ (%i10) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]); " resolvante lineaire " 4 (%o10) y - 1 (%i11) resolvante: symetrique$ (%i12) resolvante (x^4 - 1, x, x1 + x2 + x3, [x1, x2, x3]); " resolvante symetrique " 4 (%o12) y - 1 (%i13) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]); " resolvante symetrique " 6 2 (%o13) y - 4 y - 1 (%i14) resolvante: alternee$ (%i15) resolvante (x^4 + x + 1, x, x1 - x2, [x1, x2]); " resolvante alternee " 12 8 6 4 2 (%o15) y + 8 y + 26 y - 112 y + 216 y + 229 (%i16) resolvante: produit$ (%i17) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]); " resolvante produit " 35 33 29 28 27 26 (%o17) y - 7 y - 1029 y + 135 y + 7203 y - 756 y 24 23 22 21 20 + 1323 y + 352947 y - 46305 y - 2463339 y + 324135 y 19 18 17 15 - 30618 y - 453789 y - 40246444 y + 282225202 y 14 12 11 10 - 44274492 y + 155098503 y + 12252303 y + 2893401 y 9 8 7 6 - 171532242 y + 6751269 y + 2657205 y - 94517766 y 5 3 - 3720087 y + 26040609 y + 14348907 (%i18) resolvante: symetrique$ (%i19) resolvante (x^7 - 7*x + 3, x, x1*x2*x3, [x1, x2, x3]); " resolvante symetrique " 35 33 29 28 27 26 (%o19) y - 7 y - 1029 y + 135 y + 7203 y - 756 y 24 23 22 21 20 + 1323 y + 352947 y - 46305 y - 2463339 y + 324135 y 19 18 17 15 - 30618 y - 453789 y - 40246444 y + 282225202 y 14 12 11 10 - 44274492 y + 155098503 y + 12252303 y + 2893401 y 9 8 7 6 - 171532242 y + 6751269 y + 2657205 y - 94517766 y 5 3 - 3720087 y + 26040609 y + 14348907 (%i20) resolvante: cayley$ (%i21) resolvante (x^5 - 4*x^2 + x + 1, x, a, []); " resolvente de Cayley " 6 5 4 3 2 (%o21) x - 40 x + 4080 x - 92928 x + 3772160 x + 37880832 x + 93392896 @end example Pela resolvente de Cayley, os 2 @'ultimos arguments s@~ao neutros e o polin@^omio fornecido na entrada deve ser necess@'ariamente de grau 5. Veja tamb@'em : @code{resolvante_bipartite}, @code{resolvante_produit_sym}, @code{resolvante_unitaire}, @code{resolvante_alternee1}, @code{resolvante_klein}, @code{resolvante_klein3}, @code{resolvante_vierer}, @code{resolvante_diedrale}. @end deffn @deffn {Fun@,{c}@~ao} resolvante_alternee1 (@var{P}, @var{x}) calcula a transforma@,{c}@~ao de @c UMM, I THINK THE TEX STUFF SHOULD BE REPLACED BY @code @code{@var{P}(@var{x})} de grau @var{n} pela fun@,{c}@~ao $\prod_@{1\leq i