{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart: with(algcur ves):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "The puiseux program can give unnecessarily large answers. Reading the following code into Map le before using the puiseux command will help to prevent one (but not \+ all) of the causes of these large answers, namely it will help to prev ent some unnecessary expanding." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2225 "`algcurves/lift_exp` := proc(v, f, x, y)\nlocal i, ii, r, re s, v7, vv7, v3, ext, a, j, n, np, ram, j3;\n if v[5] = 1 then retur n \{v\} end if;\n v3 := degree(v[3], x);\n res := \{\};\n r : = v[1] + y*x^v[2];\n vv7 := v[7]*v3 + v[2] - 1;\n vv7 := vv7 + v [5];\n ii := `algcurves/truncate_subs`(subs(x = v[3], f), x, y, r, \+ vv7 + 1,\n v[4]);\n if ii = 0 then error \"degree estimate w as wrong\" end if;\n v7 := (ldegree(ii, x) - v[2])/v3;\n r := `a lgcurves/v_ext_m`(\n `algcurves/g_factors`(tcoeff(ii, x), y, v[ 4]), y);\n for i in r do res := res union `algcurves/lift_exp`([\n \+ v[1] + x^v[2]*i[1], v[2] + 1, v[3], [op(i[3]), op(v[4])], i[2], \n v[6]*i[4], v7, [op(v[8]), [op(1 .. 4, v)]]], f, x, y)\n e nd do;\n if add(i[5]*i[6]*degree(i[3], x)/(v[6]*v3), i = res) <>\n \+ degree(tcoeff(ii, x), y) then error \"found wrong number of expansi ons\"\n end if;\n if v[5] = degree(tcoeff(ii, x), y) then\n \+ if ldegree(ii, x) <> vv7 then error \"degree estimate was wrong\"\n end if;\n return res\n end if;\n ii := collect(ii , y);\n ii := add(`algcurves/normal_tcoeff`(coeff(ii, y, i), x)*y^i ,\n i = 0 .. degree(ii, y));\n np := `algcurves/Newtonpolygo n`(ii, x, y);\n if nops(np) = 2 and np[1][3] = 0 then\n erro r \"found wrong number of expansions\"\n end if;\n for j in np d o\n if 2 < nops(j) and 0 < j[3] and j[3] < 1 then\n \+ r := `algcurves/g_factors`(j[4], x, v[4]);\n r := `algcurve s/v_ext_m`(r, x);\n for i in r do\n j3 := j[ 3] - v[2];\n ext := [op(i[3]), op(v[4])];\n \+ n := mods(1/numer(j3), denom(j3));\n ram := i[1]^n* x^denom(j3);\n a := v[2]*denom(j3) - numer(j[3]);\n \+ res := res union `algcurves/lift_exp`([collect(\n \+ subs(x = ram, v[1])\n + x^a*i[1]^((1 - \+ n*numer(j3))/denom(j3)), x, normal),\n a + 1, norma l(subs(x = ram, v[3])), ext, i[2],\n v[6]*i[4],\n \+ (j[2] - j[1]*j[3] - a/degree(ram, x))/degree(v[3], x ),\n [op(v[8]), [op(1 .. 4, v)]]], f, x, y)\n \+ end do\n end if\n end do;\n res\nend proc:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "A:=(x^2+y)^3+a*x*y^3+b*y^4+ c*x^3*y^2+d*x^2*y^3+e*x*y^4+f*y^5+g*x^5*y+h*x^4*y^2+j*x^3*y^3+k*x^2*y^ 4+l*x*y^5+m*y^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,<*$),&*$)% \"xG\"\"#\"\"\"F-%\"yGF-\"\"$F-F-*(%\"aGF-F+F-)F.F/F-F-*&%\"bGF-)F.\" \"%F-F-*(%\"cGF-)F+F/F-)F.F,F-F-*(%\"dGF-F*F-F2F-F-*(%\"eGF-F+F-F5F-F- *&%\"fGF-)F.\"\"&F-F-*(%\"gGF-)F+FBF-F.F-F-*(%\"hGF-)F+F6F-F:F-F-*(%\" jGF-F9F-F2F-F-*(%\"kGF-F*F-F5F-F-*(%\"lGF-F+F-FAF-F-*&%\"mGF-)F.\"\"'F -F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 532 "define polynomial [1] Hav ing three segments where the slope is -2 is just flat out nasty becaus e we will have three different roots. We break this into two cases: a ll three are the same, two of the roots are the same and one is differ ent. If all three are different then the jets completely split and th ere are no more cases to consider. So, in this case, by substituting \+ in the proper values for the coefficients of the terms that are repres ented by points on the Polygon, we ensure that all are three of the ro ots are the same." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "puiseu x(A,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'RootOfG6#,**&%\" mG\"\"\")%#_ZG\"\"$F*F**&%\"fGF*)F,\"\"#F*F*F*F**&%\"bGF*F,F*F*,&*$)% \"xGF1F*!\"\"*&*&F7F*,(%\"gGF*%\"aGF*%\"cGF8F*#\"\"(F-F;!\"#F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numerator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A1:=eval(A,a=c-g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,<*$),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-\"\"$F-F-*(,&% \"cGF-%\"gG!\"\"F-F+F-)F.F/F-F-*&%\"bGF-)F.\"\"%F-F-*(F2F-)F+F/F-)F.F, F-F-*(%\"dGF-F*F-F5F-F-*(%\"eGF-F+F-F8F-F-*&%\"fGF-)F.\"\"&F-F-*(F3F-) F+FDF-F.F-F-*(%\"hGF-)F+F9F-F " 0 "" {MPLTEXT 1 0 20 "puiseux(A1,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%-%'RootOfG6#,**&%\"mG\"\"\")%#_ZG\"\"$F*F**&%\"fGF*)F,\"\"#F*F*F* F**&%\"bGF*F,F*F*,&*$)%\"xGF1F*!\"\"*(F7F-,(%\"dGF8F3F*%\"hGF*F*,&*&F1 F*%\"gGF*F*%\"cGF8F8F*,&F5F8*&)*&F7F*F=F8#\"\"&F1F*)F=F-F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim=0 [4]" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "A2:=eval(A1,c=2*g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G,<*$),&*$)%\"xG\"\"#\"\"\"F-%\"yGF-\"\"$F-F-*(F+F -)F.F/F-%\"gGF-F-*&%\"bGF-)F.\"\"%F-F-**F,F-F2F-)F+F/F-)F.F,F-F-*(%\"d GF-F*F-F1F-F-*(%\"eGF-F+F-F5F-F-*&%\"fGF-)F.\"\"&F-F-*(F2F-)F+FAF-F.F- F-*(%\"hGF-)F+F6F-F9F-F-*(%\"jGF-F8F-F1F-F-*(%\"kGF-F*F-F5F-F-*(%\"lGF -F+F-F@F-F-*&%\"mGF-)F.\"\"'F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A2,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'Roo tOfG6#,**&%\"mG\"\"\")%#_ZG\"\"$F*F**&%\"fGF*)F,\"\"#F*F*F*F**&%\"bGF* F,F*F*,&*&),(%\"dG!\"\"F3F*%\"hGF*F-F*)*&%\"xGF*,(F8F*F3F9F:F9F9#\"\") F-F*F9*(F7F1F=F1F>!\"#F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numer ator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "A3:=eval(A2,b =d-h);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A3G,<*$),&*$)%\"xG\"\"#\" \"\"F-%\"yGF-\"\"$F-F-*(F+F-)F.F/F-%\"gGF-F-*&,&%\"dGF-%\"hG!\"\"F-)F. \"\"%F-F-**F,F-F2F-)F+F/F-)F.F,F-F-*(F5F-F*F-F1F-F-*(%\"eGF-F+F-F8F-F- *&%\"fGF-)F.\"\"&F-F-*(F2F-)F+FCF-F.F-F-*(F6F-)F+F9F-F " 0 "" {MPLTEXT 1 0 20 "puiseux(A3,x=0,y,0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'RootOfG6#,*\"\"\"F(*&%\"mGF()%#_Z G\"\"$F(F(*&%\"fGF()F,\"\"#F(F(*&,&%\"dGF(%\"hG!\"\"F(F,F(F(,&*$)%\"xG F1F(F6*&)F:F-F(-F%6#,,*$F+F(F(*&F0F(%\"gGF(F6*&,&F4F6*&F1F(F5F(F(F(F,F (F(%\"jGF6%\"eGF(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim= 0 [4]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "discrim(_Z^3-_Z^2* g+(-d+2*h)*_Z-j+e,_Z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,B*(\"\"%\" \"\")%\"gG\"\"#F&)%\"hGF)F&F&**\"#OF&F(F&F+F&%\"jGF&F&**F-F&F(F&F+F&% \"eGF&!\"\"**F%F&F'F&F+F&%\"dGF&F1*&\"#FF&)F.F)F&F1*(\"#aF&F.F&F0F&F&* *\"#=F&F.F&F(F&F3F&F1*&F5F&)F0F)F&F1**F:F&F0F&F(F&F3F&F&*&F'F&)F3F)F&F &*(F%F&)F(\"\"$F&F.F&F1*(F%F&FAF&F0F&F&*(\"#[F&F3F&F*F&F&*(\"#CF&F?F&F +F&F1*&F%F&)F3FBF&F&*&\"#KF&)F+FBF&F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(%,j);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$,,*(\" \"$!\"\"%\"gG\"\"\"%\"dGF(F&*(\"\"#F(\"#FF&F'F%F&%\"eGF(**F+F(F%F&F'F( %\"hGF(F(*(F+F(F,F&,6*(F,F()F'F+F()F)F+F(F(*(\"\"*F()F'\"\"%F(F)F(F(** \"$3\"F(F3F(F/F(F)F(F&*$)F'\"\"'F(F(*(\"#=F(F7F(F/F(F&*(F:F(F3F()F/F+F (F(*&F,F()F)F%F(F(*(\"$C$F(F)F(FAF(F(*(\"$i\"F(F4F(F/F(F&*&\"$;#F()F/F %F(F&#F(F+F(,,*(F%F&F'F(F)F(F&*(F+F(F,F&F'F%F&F-F(**F+F(F%F&F'F(F/F(F( *(F+F(F,F&F1FKF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "A4:=ev al(A3,j=-1/3*g*d-2/27*g^3+e+2/3*g*h+2/27*(27*g^2*d^2+9*g^4*d-108*g^2*h *d+g^6-18*g^4*h+108*g^2*h^2+27*d^3+324*d*h^2-162*d^2*h-216*h^3)^(1/2)) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A4G,<*$),&*$)%\"xG\"\"#\"\"\"F -%\"yGF-\"\"$F-F-*(F+F-)F.F/F-%\"gGF-F-*&,&%\"dGF-%\"hG!\"\"F-)F.\"\"% F-F-**F,F-F2F-)F+F/F-)F.F,F-F-*(F5F-F*F-F1F-F-*(%\"eGF-F+F-F8F-F-*&%\" fGF-)F.\"\"&F-F-*(F2F-)F+FCF-F.F-F-*(F6F-)F+F9F-F " 0 " " {MPLTEXT 1 0 11 "factor(A4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L*$ )%\"xG\"\"'\"\"\"F(*(\"\"$F()F&\"\"%F(%\"yGF(F(*(F*F()F&\"\"#F()F-F0F( F(*$)F-F*F(F(*(F&F(F3F(%\"gGF(F(*&)F-F,F(%\"dGF(F(*&F7F(%\"hGF(!\"\"** F0F(F5F()F&F*F(F1F(F(*(F8F(F/F(F3F(F(*(%\"eGF(F&F(F7F(F(*&%\"fGF()F-\" \"&F(F(*(F5F()F&FDF(F-F(F(*(F:F(F+F(F1F(F(*,F*F;F&F*F-F*F5F(F8F(F;*,F0 F(\"#FF;F&F*F-F*F5F*F;*(F=F(F3F(F@F(F(*.F0F(F*F;F&F*F-F*F5F(F:F(F(*,F0 F(FJF;F&F*F-F**$),(*&F'F(F:F(F;*&F*F(F8F(F(*$)F5F0F(F(F*F(#F(F0F(*(%\" kGF(F/F(F7F(F(*(%\"lGF(F&F(FCF(F(*&%\"mGF()F-F'F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(-6*h+3*d+g^2-z^2,h);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*&\"\"#!\"\"%\"dG\"\"\"F(*&\"\"'F&%\"gGF%F(*&F *F&%\"zGF%F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 230 "eval(x^6+3 *x^4*y+3*x^2*y^2+y^3+x*y^3*g+y^4*d-y^4*h+2*g*x^3*y^2+d*x^2*y^3+e*x*y^4 +f*y^5+g*x^5*y+h*x^4*y^2-1/3*x^3*y^3*g*d-2/27*x^3*y^3*g^3+x^3*y^3*e+2/ 3*x^3*y^3*g*h+2/27*x^3*y^3*z^3+k*x^2*y^4+l*x*y^5+m*y^6,h=1/2*d+1/6*g^2 -1/6*z^2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L*$)%\"xG\"\"'\"\"\"F(* (\"\"$F()F&\"\"%F(%\"yGF(F(*(F*F()F&\"\"#F()F-F0F(F(*$)F-F*F(F(*(F&F(F 3F(%\"gGF(F(*&)F-F,F(%\"dGF(F(*&F7F(,(*&F0!\"\"F8F(F(*&F'F " 0 "" {MPLTEXT 1 0 271 " A4B:=x^6+3*x^4*y+3*x^2*y^2+y^3+x*y^3*g+y^4*d-y^4*(1/2*d+1/6*g^2-1/6*z^ 2)+2*g*x^3*y^2+d*x^2*y^3+e*x*y^4+f*y^5+g*x^5*y+(1/2*d+1/6*g^2-1/6*z^2) *x^4*y^2-1/3*x^3*y^3*g*d-2/27*x^3*y^3*g^3+x^3*y^3*e+2/3*x^3*y^3*g*(1/2 *d+1/6*g^2-1/6*z^2)+2/27*x^3*y^3*z^3+k*x^2*y^4+l*x*y^5+m*y^6;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%$A4BG,L*$)%\"xG\"\"'\"\"\"F**(\"\"$F *)F(\"\"%F*%\"yGF*F**(F,F*)F(\"\"#F*)F/F2F*F**$)F/F,F*F**(F(F*F5F*%\"g GF*F**&)F/F.F*%\"dGF*F**&F9F*,(*&F2!\"\"F:F*F**&F)F>F7F2F**&F)F>%\"zGF 2F>F*F>**F2F*F7F*)F(F,F*F3F*F**(F:F*F1F*F5F*F**(%\"eGF*F(F*F9F*F**&%\" fGF*)F/\"\"&F*F**(F7F*)F(FJF*F/F*F**(FF(F,F/F,F7F*F :F*F>*,F2F*\"#FF>F(F,F/F,F7F,F>*(FCF*F5F*FFF*F**.F2F*F,F>F(F,F/F,F7F*F F(F,F/F,FAF,F**(%\"kGF*F1F*F9F*F**(%\"lGF*F(F*FIF*F**&% \"mGF*)F/F)F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate pol ynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "puiseux(A4B, x=0,y,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,&*$)%\"xG\"\"#\"\"\"! \"\"*&)F'\"\"$F),&*&F-F*%\"gGF)F)*(F(F)F-F*%\"zGF)F)F)F),(**\"(cI])F*, 8*&\"#aF)%\"fGF)F)*&F8F)%\"kGF)F**$)F2\"\"%F)F**(F-F)%\"dGF))F0F(F)F** $)F0F>F)F)*(F-F)F@F))F2F(F)F***\"\"'F)F@F)F0F)F2F)F)*(\"#=F)%\"eGF)F0F )F)*(FIF)FJF)F2F)F**(F(F))F2F-F)F0F)F)*(F(F))F0F-F)F2F)F*F>F2!\"%,$**F 8F)F'F)F6F*F2F)F*#\"\"(F(F)*(F-F*,&F0F)F2F*F)F'F-F)F%F*-%'RootOfG6#,*F GF)*(FGF)%\"mGF))%#_ZGF-F)F)*(FGF)F9F))FhnF(F)F)*&,(*$FAF)F**&F-F)F@F) F)*$FEF)F)F)FhnF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numerator= 0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "solve(54*f-54*k-z^ 4-3*d*g^2+g^4-3*d*z^2+6*d*g*z+18*e*g-18*e*z+2*z^3*g-2*g^3*z,f);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,6*&\"#a!\"\"%\"gG\"\"%F&%\"kG\"\"\"*& F%F&%\"zGF(F**(\"#=F&%\"dGF*F'\"\"#F**(\"\"$F&%\"eGF*F,F*F**(F.F&F/F*F ,F0F***\"\"*F&F/F*F'F*F,F*F&*(F2F&F3F*F'F*F&*(\"#FF&F,F2F'F*F&*(F9F&F' F2F,F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "A5:=eval(A4B,f =-1/54*g^4+k+1/54*z^4+1/18*d*g^2+1/3*e*z+1/18*d*z^2-1/9*d*g*z-1/3*e*g- 1/27*z^3*g+1/27*g^3*z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A5G,L*$) %\"xG\"\"'\"\"\"F**(\"\"$F*)F(\"\"%F*%\"yGF*F**(F,F*)F(\"\"#F*)F/F2F*F **$)F/F,F*F**(F(F*F5F*%\"gGF*F**&)F/F.F*%\"dGF*F**&F9F*,(*&F2!\"\"F:F* F**&F)F>F7F2F**&F)F>%\"zGF2F>F*F>**F2F*F7F*)F(F,F*F3F*F**(F:F*F1F*F5F* F**(%\"eGF*F(F*F9F*F**&,6*&\"#aF>F7F.F>%\"kGF**&FJF>FAF.F**(\"#=F>F:F* F7F2F**(F,F>FFF*FAF*F**(FNF>F:F*FAF2F***\"\"*F>F:F*F7F*FAF*F>*(F,F>FFF *F7F*F>*(\"#FF>FAF,F7F*F>*(FUF>F7F,FAF*F*F*)F/\"\"&F*F**(F7F*)F(FXF*F/ F*F**(FF(F,F/F,F7F*F:F*F>*,F2F*FUF>F(F,F/F,F7F,F>*( FCF*F5F*FFF*F**.F2F*F,F>F(F,F/F,F7F*FF(F,F/F,FAF,F**(FK F*F1F*F9F*F**(%\"lGF*F(F*FWF*F**&%\"mGF*)F/F)F*F*" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A5,x=0,y,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,&*$)%\"xG\"\"#\"\"\"!\"\"*&)F'\"\"$F),&*&F-F*%\"gGF) F)*(F(F)F-F*%\"zGF)F)F)F),(F%F**&F,F),&*&F-F*F0F)F)*&F-F*F2F)F*F)F)*&) F'\"\"%F)-%'RootOfG6#,F*(\"$i\"F))%#_ZGF(F)F2F)F)*&,0*(\"#!*F)F0F))F2F (F)F**(\"#=F))F0F(F)F2F)F)*&F@F)%\"eGF)F)*&FIF))F0F-F)F)*&\"#aF))F2F-F )F)*(FPF)F0F)%\"dGF)F**(FPF)FSF)F2F)F)F)FBF)F)*&F@F)%\"lGF)F)*(\"\"*F) FSF)FNF)F**(FPF)F0F)%\"kGF)F***\"#FF)FSF)F0F)FGF)F**(\"\"(F))F0F:F)F2F )F**(FXF)FSF)FQF)F)*(\"#OF)FLF)FGF)F)*(F\\oF)FJF)FLF)F)*(FIF)FQF)FJF)F )*(\"# " 0 "" {MPLTEXT 1 0 216 "discrim(1 62*_Z^2*z+(-90*g*z^2+18*g^2*z+162*e+18*g^3+54*z^3-54*g*d+54*d*z)*_Z+16 2*l-9*d*g^3-54*g*k-27*d*g*z^2-7*g^4*z+9*d*z^3+36*e*z^2+36*g^2*e+18*z^3 *g^2-17*z^4*g+54*k*z+27*d*g^2*z-2*g^3*z^2+5*z^5+3*g^5-72*e*g*z,_Z);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#,P**\"%Ke\"\"\"%\"dGF&)%\"gG\"\"#F&)% \"zGF*F&!\"\"**\"&#*\\$F&F,F&F)F&%\"kGF&F&**\"&'\\F&F'F&F)F&)F,\"\"$F&F&**F2F&F3F&F)F&F+F&F&**F%F&F,F&F'F&)F)F7F& F&**F%F&)F'F*F&F,F&F)F&F-**F2F&F3F&F)F&F'F&F-*(\"'w\\5F&F,F&%\"lGF&F-* (\"%?;F&)F)\"\"%F&F+F&F&*(F%F&F3F&F6F&F-*(FBF&)F,FDF&F(F&F-*(\"%'H\"F& )F,\"\"&F&F)F&F&*(F/F&F0F&F+F&F-*(FIF&F,F&)F)FKF&F-*(\"%;HF&F " 0 "" {MPLTEXT 1 0 11 "solve(%,k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**\"$3\"!\"\",L**\"#=\"\"\"%\"dGF*)%\"gG\"\"#F*)%\"zG F.F*F&*(\"\"%F*F0F*)F-\"\"&F*F&**\"#aF*F0F*F,F*%\"eGF*F&**\"\"'F*F+F*F -F*)F0\"\"$F*F***F6F*F7F*F-F*F/F*F***F)F*F0F*F+F*)F-F;F*F***F)F*)F+F.F *F0F*F-F*F&**F6F*F7F*F-F*F+F*F&*(\"$C$F*F0F*%\"lGF*F&*(F4F*)F-F2F*F/F* F**(F)F*F7F*F:F*F&*(F4F*)F0F2F*F,F*F&*(F2F*)F0F4F*F-F*F**$)F-F9F*F**$) F0F9F*F&*(\"\"*F*F@F*F/F*F**&\"#\")F*)F7F.F*F**(FQF*F,F*F@F*F***F6F*F+ F*F0F*F7F*F**(F)F*F>F*F7F*F**(F9F*FFF*F+F*F&F*F0F&,&F-F*F0F&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A6:=eval(A5,k=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A6G,L*$)%\"xG\"\"'\"\"\"F**(\"\"$F*)F(\" \"%F*%\"yGF*F**(F,F*)F(\"\"#F*)F/F2F*F**$)F/F,F*F**(F(F*F5F*%\"gGF*F** &)F/F.F*%\"dGF*F**&F9F*,(*&F2!\"\"F:F*F**&F)F>F7F2F**&F)F>%\"zGF2F>F*F >**F2F*F7F*)F(F,F*F3F*F**(F:F*F1F*F5F*F**(%\"eGF*F(F*F9F*F**&,6*&\"#aF >F7F.F>**\"$3\"F>,L**\"#=F*F:F*)F7F2F*)FAF2F*F>*(F.F*FAF*)F7\"\"&F*F>* *FJF*FAF*FPF*FFF*F>**F)F*F:F*F7F*)FAF,F*F***FJF*FFF*F7F*FQF*F***FOF*FA F*F:F*)F7F,F*F***FOF*)F:F2F*FAF*F7F*F>**FJF*FFF*F7F*F:F*F>*(\"$C$F*FAF *%\"lGF*F>*(FTF*)F7F.F*FQF*F**(FOF*FFF*FWF*F>*(FTF*)FAF.F*FPF*F>*(F.F* )FAFTF*F7F*F**$)F7F)F*F**$)FAF)F*F>*(\"\"*F*FfnF*FQF*F**&\"#\")F*)FFF2 F*F**(FgoF*FPF*FfnF*F***FJF*F:F*FAF*FFF*F**(FOF*FZF*FFF*F**(F)F*F\\oF* F:F*F>F*FAF>,&F7F*FAF>F>F>*&FJF>FAF.F**(FOF>F:F*F7F2F**(F,F>FFF*FAF*F* *(FOF>F:F*FAF2F***FgoF>F:F*F7F*FAF*F>*(F,F>FFF*F7F*F>*(\"#FF>FAF,F7F*F >*(FgpF>F7F,FAF*F*F*)F/FTF*F**(F7F*)F(FTF*F/F*F**(F F(F,F/F,F7F*F:F*F>*,F2F*FgpF>F(F,F/F,F7F,F>*(FCF*F5F*FFF*F**.F2F*F,F>F (F,F/F,F7F*FF(F,F/F,FAF,F**.FLF>FMF*FAF>F_pF>F(F2F/F.F >*(FjnF*F(F*FipF*F**&%\"mGF*)F/F)F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A6,x=0,y,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,**,\"4K%euZ;1kYn!\"\",`s**\"$H(\"\"\")%\"gG\"\"#F+)%\"eGF.F+%\"d GF+F'*(\"#FF+)F-\"\"'F+)F1F.F+F+*(\"\")F+)F-\"\"(F+)%\"zG\"\"$F+F+*(\" #9F+F4F+)F<\"\"%F+F'*(F8F+)F-F=F+)F " 0 "" {MPLTEXT 1 0 825 "sol ve(-729*g^2*e^2*d+27*g^6*d^2+8*g^7*z^3-14*g^6*z^4-8*g^3*z^7+14*g^4*z^6 -729*e^2*z^4-972*g^3*e^2*z+108*d^3*g^3*z-9*g^8*d+27*g^7*e-108*g^5*d^2* z-135*d^2*z^4*g^2-36*g^5*d*z^3+90*g^4*d*z^4+1215*d^2*g*e*z^2-243*z^4*g ^3*e+162*z^3*g^4*e+486*g^2*e^2*z^2-4*g^9*z+81*g^5*e*z^2-648*g^3*e*d*z^ 2+108*d^3*z^3*g+108*d^2*z^5*g-2916*e^3*z-2916*l*g*d*z^2-54*z^7*e-3*z^8 *g^2+4*z^9*g-9*z^8*d+972*z^5*l-27*d^3*z^4+3*g^8*z^2-27*d^2*z^6+5832*m* z^4-z^10+972*e^2*g*z^3-2187*e^2*d*z^2+8748*l*e*z^2-972*z^3*l*g^2-972*z ^4*l*g+810*e*z^4*g*d-27*g^4*d^3+972*l*g^3*z^2+36*z^7*g*d+135*e*z^6*g-4 86*d^2*z^3*e+36*g^7*d*z-36*d*g^2*z^6-162*g^2*d^3*z^2+135*d^2*g^4*z^2-3 6*g^6*d*z^2-5832*m*z^3*g-108*g^6*e*z-324*e*z^5*d+2916*z^3*l*d-36*g^3*z ^5*d-324*z^3*g^2*e*d+648*g^4*e*d*z-972*d^2*g^2*e*z+2916*e^2*g*d*z+g^10 +243*g^3*e*d^2-162*g^5*e*d+243*g^4*e^2+729*g*e^3,l);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**\"$s*!\"\",br**\"$H(\"\"\")%\"gG\"\"#F*)%\"eGF-F *%\"dGF*F&*(\"#FF*)F,\"\"'F*)F0F-F*F**(\"\")F*)F,\"\"(F*)%\"zG\"\"$F*F **(\"#9F*F3F*)F;\"\"%F*F&*(F7F*)F,FF*)F,F@F*)F;F4F*F** (F)F*F.F*F?F*F&**F%F*FBF*F.F*F;F*F&**\"$3\"F*)F0F " 0 "" {MPLTEXT 1 0 17 "A7:=eval(A6,l=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A7G,L*$)% \"xG\"\"'\"\"\"F**(\"\"$F*)F(\"\"%F*%\"yGF*F**(F,F*)F(\"\"#F*)F/F2F*F* *$)F/F,F*F**(F(F*F5F*%\"gGF*F**&)F/F.F*%\"dGF*F**&F9F*,(*&F2!\"\"F:F*F **&F)F>F7F2F**&F)F>%\"zGF2F>F*F>**F2F*F7F*)F(F,F*F3F*F**(F:F*F1F*F5F*F **(%\"eGF*F(F*F9F*F**&,6*&\"#aF>F7F.F>**\"$3\"F>,L**\"#=F*F:F*)F7F2F*) FAF2F*F>*(F.F*FAF*)F7\"\"&F*F>**FJF*FAF*FPF*FFF*F>**F)F*F:F*F7F*)FAF,F *F***FJF*FFF*F7F*FQF*F***FOF*FAF*F:F*)F7F,F*F***FOF*)F:F2F*FAF*F7F*F>* *FJF*FFF*F7F*F:F*F>**F,F>FAF>,br**\"$H(F*FPF*)FFF2F*F:F*F>*(\"#FF*)F7F )F*FfnF*F**(\"\")F*)F7\"\"(F*FWF*F**(\"#9F*F_oF*)FAF.F*F>*(FaoF*FZF*)F AFcoF*F>*(FeoF*)F7F.F*)FAF)F*F**(F[oF*F\\oF*FfoF*F>**\"$s*F*FZF*F\\oF* FAF*F>**FLF*)F:F,F*FZF*FAF*F**(\"\"*F*)F7FaoF*F:F*F>*(F^oF*FboF*FFF*F* **FLF*FSF*FfnF*FAF*F>**\"$N\"F*FfnF*FfoF*FPF*F>**\"#OF*FSF*F:F*FWF*F>* *\"#!*F*FjoF*F:F*FfoF*F**,\"%:7F*FfnF*F7F*FFF*FQF*F***\"$V#F*FfoF*FZF* FFF*F>**\"$i\"F*FWF*FjoF*FFF*F***\"$'[F*FPF*F\\oF*FQF*F**(F.F*)F7FbpF* FAF*F>**\"#\")F*FSF*FFF*FQF*F**,\"$['F*FZF*FFF*F:F*FQF*F>**FLF*F`pF*FW F*F7F*F***FLF*FfnF*)FAFTF*F7F*F**(\"%;HF*)FFF,F*FAF*F>*(FJF*FhoF*FFF*F >*(F,F*)FAFaoF*FPF*F>*(F.F*)FAFbpF*F7F*F**(FbpF*FbrF*F:F*F>*(F^oF*F`pF *FfoF*F>*(F,F*FcpF*FQF*F**(F^oF*FfnF*F[pF*F>*(\"%KeF*%\"mGF*FfoF*F**$) FA\"#5F*F>**F^pF*F\\oF*F7F*FWF*F***\"%(=#F*F\\oF*F:F*FQF*F>*,\"$5)F*FF F*FfoF*F7F*F:F*F**(F^oF*FjoF*F`pF*F>**FipF*FhoF*F7F*F:F*F***FgpF*FFF*F [pF*F7F*F***FcqF*FfnF*FWF*FFF*F>**FipF*FboF*F:F*FAF*F***FipF*F:F*FPF*F [pF*F>**FaqF*FPF*F`pF*FQF*F>**FgpF*FfnF*FjoF*FQF*F***FipF*F_oF*F:F*FQF *F>**FjrF*F[sF*FWF*F7F*F>**FLF*F_oF*FFF*FAF*F>**\"$C$F*FFF*F\\rF*F:F*F >**FipF*FZF*F\\rF*F:F*F>*,F`tF*FWF*FPF*FFF*F:F*F>*,FiqF*FjoF*FFF*F:F*F AF*F**,F^pF*FfnF*FPF*FFF*FAF*F>*,F^rF*F\\oF*F7F*F:F*FAF*F**$)F7F^sF*F* **F_qF*FZF*FFF*FfnF*F***FaqF*FSF*FFF*F:F*F>*(F_qF*FjoF*F\\oF*F**(F[oF* F7F*F_rF*F*F*,0*(F,F*F7F*F:F*F>*$FWF*F**&FbpF*FFF*F**&FPF*FAF*F>*&F7F* FQF*F>*$FZF*F**(F,F*F:F*FAF*F*F>F**(FTF*FjoF*FQF*F**(FOF*FFF*FWF*F>*(F TF*FfoF*FPF*F>*(F.F*F\\rF*F7F*F**$F_oF*F**$F[pF*F>*(FbpF*FfnF*FQF*F**& FgqF*F\\oF*F**(FbpF*FPF*FfnF*F***FJF*F:F*FAF*FFF*F**(FOF*FZF*FFF*F**(F )F*FjoF*F:F*F>F*FAF>,&F7F*FAF>F>F>*&FJF>FAF.F**(FOF>F:F*F7F2F**(F,F>FF F*FAF*F**(FOF>F:F*FAF2F***FbpF>F:F*F7F*FAF*F>*(F,F>FFF*F7F*F>*(F^oF>FA F,F7F*F>*(F^oF>F7F,FAF*F*F*)F/FTF*F**(F7F*)F(FTF*F/F*F**(FF(F,F/F,F7F*F:F*F>*,F2F*F^oF>F(F,F/F,F7F,F>*(FCF*F5F*FFF*F**.F2 F*F,F>F(F,F/F,F7F*FF(F,F/F,FAF,F**.FLF>FMF*FAF>F`vF>F( F2F/F.F>*.F^pF>FinF*FA!\"#F\\uF>F(F*F/FTF>*&F[sF*)F/F)F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(A7);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,\"$s*!\"\",8**\"\"'\"\"\"%\"xGF*%\"yGF*)%\"zG\"\"#F *F&*&)F.\"\"$F*)F,F/F*F**()%\"gGF/F*F.F*F3F*F&*&)F6F2F*F3F*F**,F)F*F6F *F+F*F,F*F.F*F***F2F*F3F*F.F*%\"dGF*F**(\"\"*F*F3F*%\"eGF*F***F2F*F6F* F3F*F;F*F&*(F6F*F-F*F3F*F&*(\"#=F*F,F*F.F*F**(FBF*)F+F/F*F.F*F*F*,\\y* (\"#aF*F3F*)F.\"\"%F*F&**\"$i\"F*F3F*F;F*F-F*F&**FGF*F3F*F1F*F6F*F**, \"#\")F*FDF*F3F*F1F*F>F*F**,FBF*FDF*F3F*F1F*F8F*F**,\"#:F*FDF*F3F*)F. \"\"&F*F6F*F&*.FGF*FDF*F3F*F-F*F;F*F5F*F**.\"#OF*FDF*F3F*F1F*F;F*F6F*F &*.\"$3\"F*FDF*F3F*F-F*F>F*F6F*F&*,\"#7F*FDF*F3F*F-F*)F6FIF*F&*,FBF*)F ,F2F*F1F*F;F*F6F*F**,FGF*FgnF*F-F*F>F*F6F*F&*,F2F*F+F*FgnF*F6F*)F.F)F* F**,FSF*F+F*FgnF*FenF*F1F*F***F+F*FgnF*)F6FSF*F-F*F**(\"$V#F*)F>F/F*Fg nF*F**(FGF*)F+FIF*FHF*F&*,FGF*FgnF*F-F*F;F*F5F*F**(F)F*FgnF*FjnF*F&**F +F*FgnF*F5F*FRF*F&*.FZF*F+F*FgnF*F8F*F-F*F;F*F&*.FZF*F+F*FgnF*F5F*F1F* F;F*F&*.\"#FF*F+F*FgnF*F5F*F-F*F>F*F**.\"#jF*F+F*FgnF*F6F*F1F*F>F*F**. FBF*F+F*FgnF*F6F*FHF*F;F*F***F%F*%\"mGF*F-F*)F,FIF*F&*,FKF*F3F*F;F*F6F *F.F*F**,\"$C$F*F`oF*FgnF*F.F*F+F*F&*,FNF*F`oF*F+F*FgnF*F6F*F**,FioF*F >F*FgnF*F5F*F.F*F**,FBF*F>F*FgnF*FenF*F+F*F***FGF*F3F*F8F*F.F*F&**\"$' [F*F3F*F>F*F.F*F&*.\"$N\"F*F>F*F+F*FgnF*F-F*F;F*F&*,FKF*F>F*FgnF*F6F*F ;F*F&*,FNF*F>F*FgnF*F;F*F.F*F&*,FKF*F>F*F3F*F+F*F-F*F&*,FGF*F>F*F3F*F8 F*FDF*F**,FKF*F>F*F,F*)F+F2F*F-F*F&*,F%F*F>F*F,F*FDF*F.F*F&*,\"#XF*F>F *F+F*FgnF*FHF*F&*.FGF*F>F*FgnF*F5F*F+F*F;F*F&*.F[pF*F>F*FgnF*F8F*F+F*F .F*F&*.FbpF*F>F*F3F*F6F*F+F*F.F*F&*.FioF*F>F*F3F*FDF*F5F*F.F*F&*.FNF*F >F*F3F*FDF*F;F*F.F*F&*.FKF*F>F*F3F*F6F*FDF*F;F*F&*.FbpF*F>F*F,F*F`qF*F 6F*F.F*F&*0\"$*=F*F>F*FgnF*F6F*F+F*F;F*F.F*F***FKF*FboF*F-F*F;F*F&**FG F*FboF*F.F*F8F*F&**FGF*FboF*F-F*F5F*F***FGF*FboF*F1F*F6F*F***FioF*FgnF *F5F*)F;F/F*F***FBF*FgnF*FenF*F;F*F&*(FgnF*F+F*)F.\"\"(F*F&**F2F*FgnF* F.F*F]oF*F***FGF*FgnF*FbrF*F-F*F&**F2F*F3F*FDF*)F6F)F*F**,FKF*FboF*F.F *F6F*F;F*F**,F2F*FgnF*FjrF*F+F*F.F*F&*,F=F*FgnF*F+F*FbrF*F1F*F&*,F)F*F gnF*F]oF*F+F*F;F*F&*,F=F*FgnF*F8F*F+F*FbrF*F**,FBF*FgnF*F.F*F;F*F8F*F& *,F)F*FgnF*F+F*F;F*FRF*F&*,FioF*FgnF*FbrF*F.F*F6F*F**,FBF*F3F*FHF*F+F* F6F*F&*,FioF*F3F*FDF*F5F*FbrF*F**,FBF*F3F*FDF*FenF*F;F*F&*,F2F*F3F*FDF *F.F*F]oF*F&*,F2F*FDF*F3F*FHF*F5F*F**(FgnF*)F6FfrF*F+F*F**(F2F*FgnF*Fj rF*F**.FioF*F3F*FDF*FbrF*F.F*F6F*F**,FGF*F3F*FDF*FbrF*F-F*F&*,FVF*F3F* FenF*F+F*F.F*F&*,FBF*F3F*F8F*F+F*F-F*F**,FGF*F3F*F+F*F1F*F;F*F&*,FGF*F 3F*F5F*F+F*F1F*F**,FBF*F,F*F`qF*F6F*FHF*F&*,FbpF*F,F*F;F*FDF*F-F*F&*,F XF*F,F*F1F*F6F*FDF*F**,FVF*F,F*F`qF*FenF*F.F*F&*,FBF*F,F*F`qF*F8F*F-F* F**,FGF*F,F*F`qF*F5F*F1F*F**,FXF*F,F*F5F*F-F*FDF*F**,FGF*F,F*F`qF*F;F* F1F*F&*,FXF*F,F*F8F*FDF*F.F*F&*.FXF*F3F*F5F*F;F*F+F*F.F*F**.FioF*FgnF* F5F*F+F*FbrF*F.F*F&*.FBF*FgnF*FenF*F+F*F;F*F.F*F**.FioF*FgnF*F6F*F+F*F brF*F-F*F***FBF*F3F*FRF*F+F*F&**FGF*F3F*F5F*F-F*F***FXF*F,F*FHF*FDF*F& **FBF*F,F*F`qF*FRF*F&*,FSF*F+F*FgnF*F8F*FHF*F&*.FGF*F3F*F6F*F+F*F-F*F; F*F&**F)F*FDF*F3F*FjnF*F***F2F*FgnF*FRF*F6F*F***FioF*FgnF*F1F*F>F*F&** F)F*FgnF*F1F*F8F*F&**FZF*FgnF*F-F*FenF*F&**FVF*FgnF*FHF*F;F*F&*.FXF*F, F*F5F*F`qF*F;F*F.F*F**.FbpF*F,F*F;F*FDF*F.F*F6F*F**.FGF*F,F*F`qF*F6F*F ;F*F-F*F&**FQF*FgnF*FHF*F5F*F***F_oF*F`oF*FDF*F3F*F***FhpF*F>F*FboF*F. F*F&**FGF*F>F*FgnF*F8F*F*F*F.!\"#,0*(F2F*F6F*F;F*F&*$F1F*F**&F=F*F>F*F **&F5F*F.F*F&*&F6F*F-F*F&*$F8F*F**(F2F*F;F*F.F*F*F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "polynomial factors. Case A complete [5]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "4 0 0" 357 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }