{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 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 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ma ple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple O utput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 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 0 169 "`algcurves/lift_exp` := proc(v, f, x, y)\nlocal i, ii, r, res , v7, vv7, v3, ext, a, j, n, np, ram, j3;\n if v[5] = 1 then return \{v\} end if;\n v3 := degree(v[3], x);\n" }{MPLTEXT 1 0 72 " re s := \{\};\n r := v[1] + y*x^v[2];\n vv7 := v[7]*v3 + v[2] - 1; \n" }{MPLTEXT 1 0 111 " vv7 := vv7 + v[5];\n ii := `algcurves/tr uncate_subs`(subs(x = v[3], f), x, y, r, vv7 + 1,\n v[4]);\n" } {MPLTEXT 1 0 129 " if ii = 0 then error \"degree estimate was wrong \" end if;\n v7 := (ldegree(ii, x) - v[2])/v3;\n r := `algcurves /v_ext_m`(\n" }{MPLTEXT 1 0 189 " `algcurves/g_factors`(tcoeff( ii, x), y, v[4]), y);\n for i in r do res := res union `algcurves/l ift_exp`([\n v[1] + x^v[2]*i[1], v[2] + 1, v[3], [op(i[3]), op( v[4])], i[2],\n" }{MPLTEXT 1 0 134 " v[6]*i[4], v7, [op(v[8]), \+ [op(1 .. 4, v)]]], f, x, y)\n end do;\n if add(i[5]*i[6]*degree( i[3], x)/(v[6]*v3), i = res) <>\n" }{MPLTEXT 1 0 131 " degree(tcoef f(ii, x), y) then error \"found wrong number of expansions\"\n end \+ if;\n if v[5] = degree(tcoeff(ii, x), y) then\n" }{MPLTEXT 1 0 107 " if ldegree(ii, x) <> vv7 then error \"degree estimate was wro ng\"\n end if;\n return res\n" }{MPLTEXT 1 0 103 " en d if;\n ii := collect(ii, y);\n ii := add(`algcurves/normal_tcoe ff`(coeff(ii, y, i), x)*y^i,\n" }{MPLTEXT 1 0 122 " i = 0 .. de gree(ii, y));\n np := `algcurves/Newtonpolygon`(ii, x, y);\n if \+ nops(np) = 2 and np[1][3] = 0 then\n" }{MPLTEXT 1 0 80 " error \+ \"found wrong number of expansions\"\n end if;\n for j in np do \n" }{MPLTEXT 1 0 153 " 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 := `algcurves/v_ext_m`(r, x);\n" }{MPLTEXT 1 0 106 " \+ for i in r do\n j3 := j[3] - v[2];\n e xt := [op(i[3]), op(v[4])];\n" }{MPLTEXT 1 0 145 " 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" }{MPLTEXT 1 0 179 " res := res union `algcurves/lift_exp`([collec t(\n subs(x = ram, v[1])\n + x^ a*i[1]^((1 - n*numer(j3))/denom(j3)), x, normal),\n" }{MPLTEXT 1 0 173 " a + 1, normal(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" }{MPLTEXT 1 0 93 " \+ [op(v[8]), [op(1 .. 4, v)]]], f, x, y)\n end do \n end if\n" }{MPLTEXT 1 0 29 " end do;\n res\nend proc: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "A:=a*x^5+b*x^2*y+c*x*y ^2+d*y^3+e*x^3*y+f*x^2*y^2+g*x*y^3+h*y^4+j*x^4*y+k*x^3*y^2+l*x^2*y^3+m *x*y^4+n*y^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,<*&%\"aG\"\"\" )%\"xG\"\"&F(F(*(%\"bGF()%\"xG\"\"#F(%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0 F(F(*&%\"dGF()%\"yG\"\"$F(F(*(%\"eGF()%\"xGF;F(%\"yGF(F(*(%\"fGF()%\"x GF0F()%\"yGF0F(F(*(%\"gGF(%\"xGF()%\"yGF;F(F(*&%\"hGF()%\"yG\"\"%F(F(* (%\"jGF()%\"xGFPF(%\"yGF(F(*(%\"kGF()%\"xGF;F()%\"yGF0F(F(*(%\"lGF()% \"xGF0F()%\"yGF;F(F(*(%\"mGF(%\"xGF()%\"yGFPF(F(*&%\"nGF()%\"yGF+F(F( " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "define polynomial [1]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "puiseux(A,x=0,y,0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"!-%'RootOfG6#,(*&%\"nG\"\"\")%#_ ZG\"\"#F+F+*&%\"hGF+F-F+F+%\"dGF+*&%\"xGF+-F&6#,(*&%\"cGF+F-F+F+%\"bGF +*&%\"dGF+)F-F.F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim= 0 [4]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A1:=eval(A,b=c^2/4 /d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,<*&%\"aG\"\"\")%\"xG\" \"&F(F(*,\"\"%!\"\"%\"cG\"\"#%\"dGF.%\"xGF0%\"yGF(F(*(%\"cGF(%\"xGF()% \"yGF0F(F(*&%\"dGF()%\"yG\"\"$F(F(*(%\"eGF()%\"xGF=F(%\"yGF(F(*(%\"fGF ()%\"xGF0F()%\"yGF0F(F(*(%\"gGF(%\"xGF()%\"yGF=F(F(*&%\"hGF()%\"yGF-F( F(*(%\"jGF()%\"xGF-F(%\"yGF(F(*(%\"kGF()%\"xGF=F()%\"yGF0F(F(*(%\"lGF( )%\"xGF0F()%\"yGF=F(F(*(%\"mGF(%\"xGF()%\"yGF-F(F(*&%\"nGF()%\"yGF+F(F (" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A1,x=0,y,0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"!-%'RootOfG6#,(*&%\"nG\"\"\")%#_ ZG\"\"#F+F+*&%\"hGF+F-F+F+%\"dGF+,&**\"#k!\"\",***F.F+%\"gGF+)%\"cGF.F +%\"dGF+F+*(\"\")F+%\"eGF+)%\"dG\"\"$F+F+*&%\"hGF+)%\"cGFAF+F5**\"\"%F +%\"fGF+%\"cGF+)%\"dGF.F+F5F.%\"dG!\"),$**F=F+%\"xGF+,***F.F+%\"gGF+)% \"cGF.F+%\"dGF+F+*(F=F+%\"eGF+)%\"dGFAF+F+*&%\"hGF+)%\"cGFAF+F5**FGF+% \"fGF+%\"cGF+)%\"dGF.F+F5F5%\"dGFGF5#FAF.F+**F.F5%\"xGF+%\"cGF+%\"dGF5 F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numerator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "solve(2*g*c^2*d+8*e*d^3-h*c^3-4*f*c *d^2,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%!\"\",(**\"\"#\" \"\"%\"gGF*)%\"cGF)F*%\"dGF*F**(\"\")F*%\"eGF*)%\"dG\"\"$F*F**&%\"hGF* )%\"cGF4F*F&F*%\"cGF&%\"dG!\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A2:=eval(A1,f=%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#A2G,<*&%\"aG\"\"\")%\"xG\"\"&F(F(*,\"\"%!\"\"%\"cG\"\"#%\"dGF.%\"xG F0%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0F(F(*&%\"dGF()%\"yG\"\"$F(F(*(%\"eG F()%\"xGF=F(%\"yGF(F(*.F-F.,(**F0F(%\"gGF()%\"cGF0F(%\"dGF(F(*(\"\")F( %\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F.F(%\"cGF.%\"dG!\"#%\"xGF0%\"yG F0F(*(%\"gGF(%\"xGF()%\"yGF=F(F(*&%\"hGF()%\"yGF-F(F(*(%\"jGF()%\"xGF- F(%\"yGF(F(*(%\"kGF()%\"xGF=F()%\"yGF0F(F(*(%\"lGF()%\"xGF0F()%\"yGF=F (F(*(%\"mGF(%\"xGF()%\"yGF-F(F(*&%\"nGF()%\"yGF+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "reevaluate polyomial [3]" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "puiseux(A2,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%\"\"!-%'RootOfG6#,(*&%\"nG\"\"\")%#_ZG\"\"#F+F+*&%\"h GF+F-F+F+%\"dGF+,&**F.!\"\"%\"xGF+%\"cGF+%\"dGF4F4*&)%\"xGF.F+-F&6#,2* *\"#;F+%\"cGF+)%\"dG\"\"&F+)F-F.F+F+*&,(**\"\")F+%\"gGF+)%\"cGF.F+)%\" dG\"\"$F+F4*(\"#KF+%\"eGF+)%\"dGFCF+F+**FHF+%\"hGF+)%\"cGFNF+)%\"dGF.F +F+F+F-F+F+*(FPF+%\"aGF+)%\"dGFCF+F4**F.F+%\"mGF+)%\"cG\"\"%F+%\"dGF+F 4**F\\oF+%\"lGF+)%\"cGFNF+)%\"dGF.F+F+*&%\"nGF+)%\"cGFCF+F+**FHF+%\"kG F+)%\"cGF.F+)%\"dGFNF+F4**F?F+%\"jGF+%\"cGF+)%\"dGF\\oF+F+F+F+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim=0 [4]" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 125 "discrim(16*c*d^5*_Z^2+(-8*g*c^2*d^3+32*e*d^ 5+8*h*c^3*d^2)*_Z-32*a*d^5-2*m*c^4*d+4*l*c^3*d^2+n*c^5-8*k*c^2*d^3+16* j*c*d^4,_Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"#k\"\"\")%\"dG\" \"%F&,:**\"#KF&)%\"dG\"\"'F&%\"cGF&%\"aGF&F&**F)F&)%\"dG\"\"$F&)%\"cGF )F&%\"lGF&!\"\"**\"\"#F&)%\"dGF;F&)%\"cG\"\"&F&%\"mGF&F&**\"#;F&)%\"dG F@F&)%\"cGF;F&%\"jGF&F9*(%\"dGF&)%\"cGF/F&%\"nGF&F9**\"\")F&)%\"dGF)F& )%\"cGF5F&%\"kGF&F&*,FOF&)%\"dGF)F&)%\"cGF;F&%\"gGF&%\"eGF&F9*()%\"dGF ;F&)%\"cGF)F&)%\"gGF;F&F&*,F;F&%\"dGF&)%\"cGF@F&%\"gGF&%\"hGF&F9*,FOF& )%\"dGF5F&)%\"cGF5F&%\"hGF&%\"eGF&F&*&)%\"cGF/F&)%\"hGF;F&F&*(FCF&)%\" dGF/F&)%\"eGF;F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 " solve(32*d^6*c*a-4*d^3*c^4*l+2*d^2*c^5*m-16*d^5*c^2*j-d*c^6*n+8*d^4*c^ 3*k-8*d^4*c^2*g*e+d^2*c^4*g^2-2*d*c^5*g*h+8*d^3*c^3*h*e+c^6*h^2+16*d^6 *e^2,j);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#;!\"\",8**\"#K\"\" \")%\"dG\"\"'F*%\"cGF*%\"aGF*F***\"\"%F*)%\"dG\"\"$F*)%\"cGF1F*%\"lGF* F&**\"\"#F*)%\"dGF9F*)%\"cG\"\"&F*%\"mGF*F**()%\"dGF9F*)%\"cGF1F*)%\"g GF9F*F**(%\"dGF*)%\"cGF-F*%\"nGF*F&**\"\")F*)%\"dGF1F*)%\"cGF4F*%\"kGF *F**,FMF*)%\"dGF1F*)%\"cGF9F*%\"gGF*%\"eGF*F&*(F%F*)%\"dGF-F*)%\"eGF9F *F**,F9F*%\"dGF*)%\"cGF>F*%\"gGF*%\"hGF*F&*,FMF*)%\"dGF4F*)%\"cGF4F*% \"hGF*%\"eGF*F**&)%\"cGF-F*)%\"hGF9F*F*F*%\"dG!\"&%\"cG!\"#F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A3:=eval(A2,j=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A3G,<*&%\"aG\"\"\")%\"xG\"\"&F(F(*,\"\"%! \"\"%\"cG\"\"#%\"dGF.%\"xGF0%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0F(F(*&%\" dGF()%\"yG\"\"$F(F(*(%\"eGF()%\"xGF=F(%\"yGF(F(*.F-F.,(**F0F(%\"gGF()% \"cGF0F(%\"dGF(F(*(\"\")F(%\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F.F(% \"cGF.%\"dG!\"#%\"xGF0%\"yGF0F(*(%\"gGF(%\"xGF()%\"yGF=F(F(*&%\"hGF()% \"yGF-F(F(*.\"#;F.,8**\"#KF()%\"dG\"\"'F(%\"cGF(%\"aGF(F(**F-F()%\"dGF =F()%\"cGF-F(%\"lGF(F.**F0F()%\"dGF0F()%\"cGF+F(%\"mGF(F(*()%\"dGF0F() %\"cGF-F()%\"gGF0F(F(*(%\"dGF()%\"cGFboF(%\"nGF(F.**FKF()%\"dGF-F()%\" cGF=F(%\"kGF(F(*,FKF()%\"dGF-F()%\"cGF0F(%\"gGF(%\"eGF(F.*(F\\oF()%\"d GFboF()%\"eGF0F(F(*,F0F(%\"dGF()%\"cGF+F(%\"gGF(%\"hGF(F.*,FKF()%\"dGF =F()%\"cGF=F(%\"hGF(%\"eGF(F(*&)%\"cGFboF()%\"hGF0F(F(F(%\"dG!\"&%\"cG FU%\"xGF-%\"yGF(F(*(%\"kGF()%\"xGF=F()%\"yGF0F(F(*(%\"lGF()%\"xGF0F()% \"yGF=F(F(*(%\"mGF(%\"xGF()%\"yGF-F(F(*&%\"nGF()%\"yGF+F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A3,x=0,y,0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%\"\"!-%'RootOfG6#,(*&%\"nG\"\"\")%#_ ZG\"\"#F+F+*&%\"hGF+F-F+F+%\"dGF+,(*,\"&oF$!\"\",R**F.F+)%\"cG\"\"&F+) %\"gG\"\"$F+)%\"dGF=F+F+*(F=F+)%\"cG\"\")F+)%\"hGF=F+F5*.\"#SF+)%\"dG \"\"%F+)%\"cGFJF+%\"gGF+%\"eGF+%\"hGF+F+*,\"#KF+)%\"dG\"\"(F+%\"cGF+% \"gGF+)%\"eGF.F+F+*,\"#;F+)%\"dGF:F+)%\"cGF=F+)%\"gGF.F+%\"eGF+F5*,FTF +)%\"cG\"\"'F+)%\"gGF.F+%\"hGF+)%\"dGF.F+F5*,FCF+)%\"cGFTF+%\"gGF+)%\" hGF.F+%\"dGF+F+*,\"#[F+)%\"dGF_oF+)%\"cGF.F+)%\"eGF.F+%\"hGF+F5*,\"#CF +)%\"cGF:F+%\"eGF+)%\"hGF.F+)%\"dGF=F+F5*,FQF+%\"kGF+)%\"dGFTF+)%\"cGF .F+%\"eGF+F5*,FCF+%\"kGF+)%\"dGF:F+)%\"cGFJF+%\"gGF+F+*,FCF+%\"kGF+)% \"dGFJF+)%\"cGF:F+%\"hGF+F5*,FfpF+%\"mGF+)%\"dGF:F+)%\"cGFJF+%\"eGF+F5 *,F_oF+%\"mGF+)%\"cGF_oF+%\"gGF+)%\"dGF=F+F+*,F_oF+%\"mGF+)%\"cGFTF+% \"hGF+)%\"dGF.F+F5**\"$G\"F+)%\"dG\"\"*F+%\"aGF+%\"eGF+F+*,FQF+)%\"dGF TF+)%\"cGF.F+%\"aGF+%\"gGF+F5*,FQF+)%\"dGF_oF+)%\"cGF=F+%\"aGF+%\"hGF+ F+*,FQF+)%\"dGF_oF+)%\"cGF=F+%\"lGF+%\"eGF+F+*,FCF+)%\"dGFJF+)%\"cGF:F +%\"lGF+%\"gGF+F5*,FCF+)%\"cGF_oF+%\"lGF+%\"hGF+)%\"dGF=F+F+*,FZF+)%\" dGFJF+)%\"cGF:F+%\"nGF+%\"eGF+F+*,FJF+)%\"cGFTF+%\"nGF+%\"gGF+)%\"dGF. F+F5*,FJF+)%\"cGFCF+%\"nGF+%\"hGF+%\"dGF+F+F=%\"cG!\"*%\"dG!#C,$*,FQF+ %\"xGF+,R**F.F+)%\"cGF:F+)%\"gGF=F+)%\"dGF=F+F+*(F=F+)%\"cGFCF+)%\"hGF =F+F5*.FGF+)%\"dGFJF+)%\"cGFJF+%\"gGF+%\"eGF+%\"hGF+F+*,FQF+)%\"dGFTF+ %\"cGF+%\"gGF+)%\"eGF.F+F+*,FZF+)%\"dGF:F+)%\"cGF=F+)%\"gGF.F+%\"eGF+F 5*,FTF+)%\"cGF_oF+)%\"gGF.F+%\"hGF+)%\"dGF.F+F5*,FCF+)%\"cGFTF+%\"gGF+ )%\"hGF.F+%\"dGF+F+*,F]pF+)%\"dGF_oF+)%\"cGF.F+)%\"eGF.F+%\"hGF+F5*,Ff pF+)%\"cGF:F+%\"eGF+)%\"hGF.F+)%\"dGF=F+F5*,FQF+%\"kGF+)%\"dGFTF+)%\"c GF.F+%\"eGF+F5*,FCF+%\"kGF+)%\"dGF:F+)%\"cGFJF+%\"gGF+F+*,FCF+%\"kGF+) %\"dGFJF+)%\"cGF:F+%\"hGF+F5*,FfpF+%\"mGF+)%\"dGF:F+)%\"cGFJF+%\"eGF+F 5*,F_oF+%\"mGF+)%\"cGF_oF+%\"gGF+)%\"dGF=F+F+*,F_oF+%\"mGF+)%\"cGFTF+% \"hGF+)%\"dGF.F+F5**FisF+)%\"dGF\\tF+%\"aGF+%\"eGF+F+*,FQF+)%\"dGFTF+) %\"cGF.F+%\"aGF+%\"gGF+F5*,FQF+)%\"dGF_oF+)%\"cGF=F+%\"aGF+%\"hGF+F+*, FQF+)%\"dGF_oF+)%\"cGF=F+%\"lGF+%\"eGF+F+*,FCF+)%\"dGFJF+)%\"cGF:F+%\" lGF+%\"gGF+F5*,FCF+)%\"cGF_oF+%\"lGF+%\"hGF+)%\"dGF=F+F+*,FZF+)%\"dGFJ F+)%\"cGF:F+%\"nGF+%\"eGF+F+*,FJF+)%\"cGFTF+%\"nGF+%\"gGF+)%\"dGF.F+F5 *,FJF+)%\"cGFCF+%\"nGF+%\"hGF+%\"dGF+F+F5%\"cGF=%\"dGFCF5#F:F.F5*,FJF5 ,(*(FJF+%\"eGF+)%\"dGF=F+F+*(%\"gGF+)%\"cGF.F+%\"dGF+F5*&%\"hGF+)%\"cG F=F+F+F+%\"cGF5%\"dG!\"$%\"xGF.F5**F.F5%\"xGF+%\"cGF+%\"dGF5F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "numerator=0 [2]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 357 "factor(2*c^5*g^3*d^3-3*c^8*h^3+40*d^4*c^ 4*g*e*h+32*d^7*c*g*e^2-16*d^5*c^3*g^2*e-7*c^6*g^2*h*d^2+8*c^7*g*h^2*d- 48*d^6*c^2*e^2*h-24*c^5*e*h^2*d^3-32*k*d^7*c^2*e+8*k*d^5*c^4*g-8*k*d^4 *c^5*h-24*m*d^5*c^4*e+6*m*c^6*g*d^3-6*m*c^7*h*d^2+128*d^9*a*e-32*d^7*c ^2*a*g+32*d^6*c^3*a*h+32*d^6*c^3*l*e-8*d^4*c^5*l*g+8*c^6*l*h*d^3+16*d^ 4*c^5*n*e-4*c^7*n*g*d^2+4*c^8*n*h*d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*(\"\"%\"\"\"%\"eGF')%\"dG\"\"$F'F'*(%\"gGF')%\"cG\"\"#F'%\"dGF '!\"\"*&%\"hGF')%\"cGF+F'F'F',6*,\"\")F')%\"dGF&F'%\"cGF'%\"gGF'%\"eGF 'F'*,\"#7F')%\"dGF+F')%\"cGF0F'%\"hGF'%\"eGF'F2*(\"#KF')%\"dG\"\"'F'% \"aGF'F'**F9F')%\"dGF&F')%\"cGF0F'%\"kGF'F2**F9F')%\"dGF+F')%\"cGF+F'% \"lGF'F'**F0F')%\"dGF0F')%\"cGF+F')%\"gGF0F'F2**FKF')%\"dGF0F')%\"cGF& F'%\"mGF'F2**F&F'%\"dGF')%\"cG\"\"&F'%\"nGF'F'*,FdoF'%\"dGF')%\"cGF&F' %\"gGF'%\"hGF'F'*(F+F')%\"cGFdoF')%\"hGF0F'F2F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "case B: short factor=0; case A: long factor=0 [7]" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(4*e*d^3-g*c^2*d+h*c^ 3,g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*(\"\"%\"\"\"%\"eGF')%\"d G\"\"$F'F'*&%\"hGF')%\"cGF+F'F'F'%\"cG!\"#%\"dG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B:=eval(A3,g=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"BG,<*&%\"aG\"\"\")%\"xG\"\"&F(F(*,\"\"%!\"\"%\"cG\" \"#%\"dGF.%\"xGF0%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0F(F(*&%\"dGF()%\"yG \"\"$F(F(*(%\"eGF()%\"xGF=F(%\"yGF(F(*.F-F.,&*(\"#;F(%\"eGF()%\"dGF=F( F(*&%\"hGF()%\"cGF=F(F(F(%\"cGF.%\"dG!\"#%\"xGF0%\"yGF0F(*,,&*(F-F(%\" eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F(F(%\"cGFP%\"dGF.%\"xGF(%\"yGF=F(* &%\"hGF()%\"yGF-F(F(*.FFF.,8**\"#KF()%\"dG\"\"'F(%\"cGF(%\"aGF(F(**F-F ()%\"dGF=F()%\"cGF-F(%\"lGF(F.**F0F()%\"dGF0F()%\"cGF+F(%\"mGF(F(*$),& *(F-F(%\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F(F0F(F(*(%\"dGF()%\"cGFeo F(%\"nGF(F.**\"\")F()%\"dGF-F()%\"cGF=F(%\"kGF(F(**FeqF()%\"dGF=F(,&*( F-F(%\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F(F(%\"eGF(F.*(FFF()%\"dGFeo F()%\"eGF0F(F(**F0F()%\"cGF=F(,&*(F-F(%\"eGF()%\"dGF=F(F(*&%\"hGF()%\" cGF=F(F(F(%\"hGF(F.*,FeqF()%\"dGF=F()%\"cGF=F(%\"hGF(%\"eGF(F(*&)%\"cG FeoF()%\"hGF0F(F(F(%\"dG!\"&%\"cGFP%\"xGF-%\"yGF(F(*(%\"kGF()%\"xGF=F( )%\"yGF0F(F(*(%\"lGF()%\"xGF0F()%\"yGF=F(F(*(%\"mGF(%\"xGF()%\"yGF-F(F (*&%\"nGF()%\"yGF+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevalu ate polynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "facto r(B);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,\"#;!\"\",&*&%\"cG\"\"\"% \"xGF*F**(\"\"#F*%\"yGF*%\"dGF*F*F*,D*,F%F*%\"aGF*)%\"dG\"\"%F*%\"cGF* )%\"xGF5F*F***)%\"xG\"\"$F*%\"yGF*%\"nGF*)%\"cG\"\"&F*F&*.F-F*)%\"xGF< 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^10*g*h*d^2-8*l*c^8*g*h*d^4-512*d^12*a*e^2,_Z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$**\"#k\"\"\")%\"dG\"#5F&)%\"cG\"\"&F&,fo*,\"$c#F&%\"h GF&)%\"dG\"\"*F&)%\"eG\"\"$F&)%\"cG\"\"#F&F&*,\"$O$F&)%\"hGF9F&)%\"dG \"\"'F&)%\"cGF,F&)%\"eGF9F&F&*,\"$%QF&%\"mGF&)%\"dG\"\")F&)%\"cG\"\"%F &)%\"eGF9F&F&*,\"#;F&%\"lGF&)%\"cG\"\"(F&)%\"gGF9F&)%\"dGF,F&!\"\"*,\" $G\"F&)%\"dGF@F&%\"aGF&)%\"cGF@F&)%\"hGF9F&F&*,FfnF&)%\"dGFJF&%\"aGF&) %\"cGFMF&)%\"gGF9F&F&*,FFF&%\"nGF&)%\"dGFUF&)%\"cGF,F&)%\"eGF9F&FZ*,F/ F&%\"lGF&)%\"dGF3F&)%\"cGF6F&)%\"eGF9F&FZ*.FfnF&%\"lGF&)%\"dGFUF&)%\"c GF,F&%\"eGF&%\"gGF&F&*.FfnF&%\"lGF&)%\"dGF@F&)%\"cGF@F&%\"eGF&%\"hGF&F Z*.\"%C5F&)%\"dGF)F&%\"aGF&%\"eGF&%\"gGF&)%\"cGF9F&FZ*.FgqF&)%\"dGF3F& %\"aGF&)%\"cGF6F&%\"eGF&%\"hGF&F&*.F/F&)%\"dGFUF&%\"aGF&)%\"cGF,F&%\"g GF&%\"hGF&FZ*(\"#8F&)%\"hGFMF&)%\"cG\"#6F&F&*,\"#CF&%\"hGF&)%\"cGFJF&) %\"gGF6F&)%\"dGF6F&FZ*.FFF&%\"hGF&)%\"dGFUF&)%\"cGFMF&)%\"eGF9F&%\"gGF &FZ*.\"$w\"F&%\"hGF&)%\"cGF@F&%\"eGF&)%\"gGF9F&)%\"dGF,F&F&*.\"$#>F&% 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1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"# k\"\"\")%\"dG\"#5F&)%\"cG\"\"&F&),(*(\"\"%F&%\"eGF&)%\"dG\"\"$F&F&*(% \"gGF&)%\"cG\"\"#F&%\"dGF&!\"\"*&%\"hGF&)%\"cGF4F&F&F9F&,2*,\"#;F&)%\" dGF4F&)%\"cGF9F&%\"hGF&%\"eGF&F&*(\"$G\"F&)%\"dG\"\"'F&%\"aGF&F&**FBF& )%\"dGF4F&)%\"cGF4F&%\"lGF&F;**\"#CF&)%\"dGF9F&)%\"cGF0F&%\"mGF&F&**F0 F&)%\"dGF9F&)%\"cGF4F&)%\"gGF9F&F&*,FBF&%\"dGF&)%\"cGF0F&%\"gGF&%\"hGF &F;**FVF&%\"dGF&)%\"cGF,F&%\"nGF&F;*(\"#8F&)%\"cGF,F&)%\"hGF9F&F&F&F& " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The short factor is the same \+ as in case B so it need not be considered [7]." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 109 "solve(16*d^3*c^2*h*e+128*d^6*a-16*d^3*c^3*l+2 4*d^2*c^4*m+4*d^2*c^3*g^2-16*d*c^4*g*h-24*d*c^5*n+13*c^5*h^2,l);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#;!\"\",0*,F%\"\"\")%\"dG\"\"$F) )%\"cG\"\"#F)%\"hGF)%\"eGF)F)*(\"$G\"F))%\"dG\"\"'F)%\"aGF)F)**\"#CF)% \"dGF))%\"cG\"\"&F)%\"nGF)F&**F9F))%\"dGF/F))%\"cG\"\"%F)%\"mGF)F)**FD F))%\"dGF/F))%\"cGF,F))%\"gGF/F)F)*,F%F)%\"dGF))%\"cGFDF)%\"gGF)%\"hGF )F&*(\"#8F))%\"cGF=F))%\"hGF/F)F)F)%\"cG!\"$%\"dGFZF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A5:=eval(A4,l=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A5G,<*&%\"aG\"\"\")%\"xG\"\"&F(F(*,\"\"%!\"\"%\"c G\"\"#%\"dGF.%\"xGF0%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0F(F(*&%\"dGF()%\" yG\"\"$F(F(*(%\"eGF()%\"xGF=F(%\"yGF(F(*.F-F.,(**F0F(%\"gGF()%\"cGF0F( %\"dGF(F(*(\"\")F(%\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F.F(%\"cGF.%\" dG!\"#%\"xGF0%\"yGF0F(*(%\"gGF(%\"xGF()%\"yGF=F(F(*&%\"hGF()%\"yGF-F(F (*.\"#;F.,8**\"#KF()%\"dG\"\"'F(%\"cGF(%\"aGF(F(*(F-F.%\"cGF(,0*,F\\oF ()%\"dGF=F()%\"cGF0F(%\"hGF(%\"eGF(F(*(\"$G\"F()%\"dGFboF(%\"aGF(F(** \"#CF(%\"dGF()%\"cGF+F(%\"nGF(F.**FepF()%\"dGF0F()%\"cGF-F(%\"mGF(F(** F-F()%\"dGF0F()%\"cGF=F()%\"gGF0F(F(*,F\\oF(%\"dGF()%\"cGF-F(%\"gGF(% \"hGF(F.*(\"#8F()%\"cGF+F()%\"hGF0F(F(F(F.**F0F()%\"dGF0F()%\"cGF+F(% \"mGF(F(*()%\"dGF0F()%\"cGF-F()%\"gGF0F(F(*(%\"dGF()%\"cGFboF(%\"nGF(F .*&%\"cGF(,0*,FKF()%\"dGF-F(%\"cGF(%\"gGF(%\"eGF(F(*,F-F()%\"dGF=F()% \"cGF0F(%\"hGF(%\"eGF(F.*(\"#'*F()%\"dGFboF(%\"aGF(F(**FKF(%\"dGF()%\" cGF+F(%\"nGF(F.**FboF()%\"dGF0F()%\"cGF-F(%\"mGF(F(*,F=F(%\"dGF()%\"cG F-F(%\"gGF(%\"hGF(F.**\"\"(F(F0F.%\"cGF+%\"hGF0F(F(F(*,FKF()%\"dGF-F() %\"cGF0F(%\"gGF(%\"eGF(F.*(F\\oF()%\"dGFboF()%\"eGF0F(F(*,F0F(%\"dGF() %\"cGF+F(%\"gGF(%\"hGF(F.*,FKF()%\"dGF=F()%\"cGF=F(%\"hGF(%\"eGF(F(*&) %\"cGFboF()%\"hGF0F(F(F(%\"dG!\"&%\"cGFU%\"xGF-%\"yGF(F(*.FKF.,0*,FKF( )%\"dGF-F(%\"cGF(%\"gGF(%\"eGF(F(*,F-F()%\"dGF=F()%\"cGF0F(%\"hGF(%\"e GF(F.*(FftF()%\"dGFboF(%\"aGF(F(**FKF(%\"dGF()%\"cGF+F(%\"nGF(F.**FboF ()%\"dGF0F()%\"cGF-F(%\"mGF(F(*,F=F(%\"dGF()%\"cGF-F(%\"gGF(%\"hGF(F.* *F\\vF(F0F.%\"cGF+%\"hGF0F(F(%\"dG!\"%%\"cGFU%\"xGF=%\"yGF0F(*.F\\oF., 0*,F\\oF()%\"dGF=F()%\"cGF0F(%\"hGF(%\"eGF(F(*(F`pF()%\"dGFboF(%\"aGF( F(**FepF(%\"dGF()%\"cGF+F(%\"nGF(F.**FepF()%\"dGF0F()%\"cGF-F(%\"mGF(F (**F-F()%\"dGF0F()%\"cGF=F()%\"gGF0F(F(*,F\\oF(%\"dGF()%\"cGF-F(%\"gGF 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F+)%\"eGF.F+)%\"hGF@F+)%\"dGFcoF+F5*,FbzF+)%\"cGFGF+%\"eGF+)%\"hGF]sF+ )%\"dGF@F+F5*,F.F+)%\"cGFSF+)%\"gGF]sF+%\"hGF+)%\"dGF]sF+F5*,FGF+)%\"c GFe[lF+%\"gGF+)%\"hGF]sF+%\"dGF+F+**F[\\lF+)%\"dGF8F+%\"aGF+)%\"eGF@F+ F+*,F]sF+)%\"cGFe[lF+%\"mGF+)%\"hGF@F+)%\"dGF.F+F5*,F]sF+)%\"cGFSF+%\" mGF+)%\"gGF@F+)%\"dGFYF+F+*.FcsF+%\"nGF+)%\"cGFDF+%\"eGF+)%\"gGF.F+)% \"dGFcoF+F+*.FcsF+%\"nGF+)%\"cGFGF+%\"eGF+)%\"hGF.F+)%\"dGF]sF+F+*.Fd^ lF+%\"nGF+)%\"cGF;F+)%\"gGF.F+%\"hGF+)%\"dGF@F+F+*.Fd^lF+%\"nGF+)%\"cG Fe[lF+%\"gGF+)%\"hGF.F+)%\"dGF.F+F5*.Fg_lF+)%\"cGFDF+%\"eGF+)%\"hGF.F+ )%\"gGF.F+)%\"dGFYF+F5*.Fd^lF+)%\"cGF\\qF+%\"eGF+%\"hGF+)%\"gGF@F+)%\" dGFcoF+F+*.FcsF+)%\"cGFSF+%\"eGF+)%\"hGF@F+%\"gGF+)%\"dGF]sF+F+*.FeoF+ )%\"dGFe[lF+%\"aGF+)%\"eGF.F+%\"gGF+)%\"cGF.F+F5*.FfwF+)%\"cGFSF+%\"mG F+%\"eGF+)%\"hGF.F+)%\"dGFYF+F5*.F;F+)%\"cGF;F+%\"mGF+%\"gGF+)%\"hGF.F +)%\"dGF@F+F+*.F;F+)%\"cGFGF+%\"mGF+)%\"gGF.F+%\"hGF+)%\"dGF]sF+F5*0Fc sF+)%\"cGFDF+%\"mGF+%\"eGF+%\"gGF+%\"hGF+)%\"dGFcoF+F+*,FOF+)%\"dGFDF+ )%\"cGFYF+)%\"eGF@F+)%\"hGF.F+F5*,F[oF+)%\"dGFDF+%\"aGF+)%\"cGFcoF+)% \"gGF@F+F5*,FbelF+)%\"dGFGF+)%\"cGF]sF+%\"mGF+)%\"eGF@F+F5F5%\"cGFbq% \"dGF;F5#FbqF.F+*,\"#;F5%\"xGF@,,*,F;F+%\"hGF+%\"eGF+%\"cGF+)%\"dGF@F+ F5*,FYF+%\"hGF+%\"gGF+)%\"cGF@F+%\"dGF+F+**F\\qF+%\"gGF+%\"eGF+)%\"dGF ]sF+F+**F.F+)%\"gGF.F+)%\"cGF.F+)%\"dGF.F+F5*(F@F+)%\"hGF.F+)%\"cGF]sF +F5F+%\"cGF5%\"dG!\"&F+*,F]sF5,(*(F]sF+%\"eGF+)%\"dGF@F+F+*(%\"gGF+)% \"cGF.F+%\"dGF+F5*&%\"hGF+)%\"cGF@F+F+F+%\"cGF5%\"dG!\"$%\"xGF.F5**F.F 5%\"xGF+%\"cGF+%\"dGF5F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "numer ator=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 853 "factor(-15*c^12* g^2*h^3*d^2+9*c^11*g^3*h^2*d^3-192*n*c^10*e*h*g*d^5-3*c^14*h^5+64*a*c^ 9*h^3*d^6+3072*d^12*a*e^2*h*c^3+512*n*c^5*d^10*e^3+384*n*c^8*e^2*h*d^7 -384*n*c^7*d^8*e^2*g+240*c^7*e^2*h^2*g*d^7+128*d^10*c^4*e^3*h*g-96*d^8 *c^6*e^2*h*g^2+768*d^9*a*c^6*e*h^2-192*a*c^8*g*h^2*d^7+192*d^8*a*c^7*g ^2*h+768*d^11*a*c^4*e*g^2-1536*d^10*a*c^5*e*g*h-192*d^8*c^7*m*e^2*h-48 *c^8*m*e*g^2*d^7+192*d^9*c^6*m*e^2*g+8*n*c^14*h^3*d-8*n*c^11*g^3*d^4-1 44*c^8*e^2*h^3*d^6-36*c^11*e*h^4*d^3-2*c^10*g^4*h*d^4+11*c^13*g*h^4*d+ 4096*d^15*a*e^3-4*c^13*m*h^3*d^2+4*c^10*m*g^3*d^5+96*n*c^9*e*g^2*d^6+9 6*n*c^11*e*h^2*d^4+24*n*c^12*g^2*h*d^3-24*n*c^13*g*h^2*d^2-84*c^9*e*h^ 2*g^2*d^5+24*c^8*e*h*g^3*d^6+96*c^10*e*h^3*g*d^4-3072*d^13*a*e^2*g*c^2 -48*c^10*m*e*h^2*d^5+12*c^12*m*g*h^2*d^3-12*c^11*m*g^2*h*d^4+96*c^9*m* e*g*h*d^6-192*d^9*c^5*e^3*h^2-64*d^9*a*c^6*g^3-256*d^11*c^4*m*e^3);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&),(*(\"\"%\"\"\"%\"eGF()%\"dG\"\"$F (F(*(%\"gGF()%\"cG\"\"#F(%\"dGF(!\"\"*&%\"hGF()%\"cGF,F(F(F,F(,,*(\"#k F()%\"dG\"\"'F(%\"aGF(F(**F'F()%\"dGF1F()%\"cGF'F(%\"mGF(F3**\"\")F(% \"dGF()%\"cG\"\"&F(%\"nGF(F(*,F1F(%\"dGF()%\"cGF'F(%\"gGF(%\"hGF(F(*(F ,F()%\"cGFJF()%\"hGF1F(F3F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Ag ain, the shorter factor is the same as in case B [7]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(64*d^6*a-4*d^2*c^4*m+8*d*c^5* n+2*d*c^4*g*h-3*c^5*h^2,m);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\" \"%!\"\",**(\"#k\"\"\")%\"dG\"\"'F*%\"aGF*F***\"\")F*%\"dGF*)%\"cG\"\" &F*%\"nGF*F**,\"\"#F*%\"dGF*)%\"cGF%F*%\"gGF*%\"hGF*F**(\"\"$F*)%\"cGF 4F*)%\"hGF7F*F&F*%\"dG!\"#%\"cG!\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A6:=eval(A5,m=%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# >%#A6G,<*&%\"aG\"\"\")%\"xG\"\"&F(F(*,\"\"%!\"\"%\"cG\"\"#%\"dGF.%\"xG F0%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0F(F(*&%\"dGF()%\"yG\"\"$F(F(*(%\"eG F()%\"xGF=F(%\"yGF(F(*.F-F.,(**F0F(%\"gGF()%\"cGF0F(%\"dGF(F(*(\"\")F( %\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F(F.F(%\"cGF.%\"dG!\"#%\"xGF0%\"yG F0F(*(%\"gGF(%\"xGF()%\"yGF=F(F(*&%\"hGF()%\"yGF-F(F(*.\"#;F.,8**\"#KF ()%\"dG\"\"'F(%\"cGF(%\"aGF(F(*(F-F.%\"cGF(,.*,F\\oF()%\"dGF=F()%\"cGF 0F(%\"hGF(%\"eGF(F(*(\"$7&F()%\"dGFboF(%\"aGF(F(**\"#CF(%\"dGF()%\"cGF +F(%\"nGF(F(*,F-F(%\"dGF()%\"cGF-F(%\"gGF(%\"hGF(F.*(F+F()%\"cGF+F()% \"hGF0F(F.**F-F()%\"dGF0F()%\"cGF=F()%\"gGF0F(F(F(F.*(F0F.%\"cGF(,**( \"#kF()%\"dGFboF(%\"aGF(F(**FKF(%\"dGF()%\"cGF+F(%\"nGF(F(*,F0F(%\"dGF ()%\"cGF-F(%\"gGF(%\"hGF(F(*(F=F()%\"cGF+F()%\"hGF0F(F.F(F(*()%\"dGF0F ()%\"cGF-F()%\"gGF0F(F(*(%\"dGF()%\"cGFboF(%\"nGF(F.*&%\"cGF(,,*,FKF() %\"dGF-F(%\"cGF(%\"gGF(%\"eGF(F(*,F-F()%\"dGF=F()%\"cGF0F(%\"hGF(%\"eG F(F.*(\"$#>F()%\"dGFboF(%\"aGF(F(**F-F(%\"dGF()%\"cGF+F(%\"nGF(F(*&)% \"cGF+F()%\"hGF0F(F.F(F(*,FKF()%\"dGF-F()%\"cGF0F(%\"gGF(%\"eGF(F.*(F \\oF()%\"dGFboF()%\"eGF0F(F(*,F0F(%\"dGF()%\"cGF+F(%\"gGF(%\"hGF(F.*,F KF()%\"dGF=F()%\"cGF=F(%\"hGF(%\"eGF(F(*&)%\"cGFboF()%\"hGF0F(F(F(%\"d G!\"&%\"cGFU%\"xGF-%\"yGF(F(*.FKF.,,*,FKF()%\"dGF-F(%\"cGF(%\"gGF(%\"e GF(F(*,F-F()%\"dGF=F()%\"cGF0F(%\"hGF(%\"eGF(F.*(FauF()%\"dGFboF(%\"aG F(F(**F-F(%\"dGF()%\"cGF+F(%\"nGF(F(*&)%\"cGF+F()%\"hGF0F(F.F(%\"dG!\" %%\"cGFU%\"xGF=%\"yGF0F(*.F\\oF.,.*,F\\oF()%\"dGF=F()%\"cGF0F(%\"hGF(% \"eGF(F(*(F`pF()%\"dGFboF(%\"aGF(F(**FepF(%\"dGF()%\"cGF+F(%\"nGF(F(*, F-F(%\"dGF()%\"cGF-F(%\"gGF(%\"hGF(F.*(F+F()%\"cGF+F()%\"hGF0F(F.**F-F ()%\"dGF0F()%\"cGF=F()%\"gGF0F(F(F(%\"cG!\"$%\"dGFi\\l%\"xGF0%\"yGF=F( *.F-F.,**(F`rF()%\"dGFboF(%\"aGF(F(**FKF(%\"dGF()%\"cGF+F(%\"nGF(F(*,F 0F(%\"dGF()%\"cGF-F(%\"gGF(%\"hGF(F(*(F=F()%\"cGF+F()%\"hGF0F(F.F(%\"d GFU%\"cGF`z%\"xGF(%\"yGF-F(*&%\"nGF()%\"yGF+F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A6,x=0,y,0);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%\"\"!-%'RootOfG6#,(*&%\"nG\"\"\")%#_ZG\"\"#F+F+*&%\"h GF+F-F+F+%\"dGF+,***F.!\"\"%\"xGF+%\"cGF+%\"dGF4F4*,\"\"%F4%\"xGF.,(*( F9F+%\"eGF+)%\"dG\"\"$F+F4*(%\"gGF+)%\"cGF.F+%\"dGF+F+*&%\"hGF+)%\"cGF @F+F4F+%\"dG!\"$%\"cGF4F+*,\"#;F4%\"xGF@,,*,\"#7F+%\"hGF+%\"eGF+%\"cGF +)%\"dGF@F+F4*,\"\"&F+%\"hGF+%\"gGF+)%\"cGF@F+%\"dGF+F+**\"\")F+%\"gGF +%\"eGF+)%\"dGF9F+F+**F.F+)%\"gGF.F+)%\"cGF.F+)%\"dGF.F+F4*(F@F+)%\"hG F.F+)%\"cGF9F+F4F+%\"cGF4%\"dG!\"&F+*&)%\"xGF9F+-F&6#,fq**FNF+)%\"cG\" #6F+)%\"gG\"\"'F+)%\"dGFjpF+F+*,FNF+%\"nGF+)%\"cG\"#8F+)%\"gGF9F+)%\"d GFYF+F+*,FNF+%\"nGF+)%\"cG\"#)F+)%\"dGFgpF+)%\"cGFjrF+%\"eGF+%\"hGF+%\"gGF+F4*,\"%cmF+)%\"dGFevF +)%\"cGFevF+%\"eGF+)%\"hGF.F+F+*,\"%/BF+)%\"cGFgpF+)%\"gGF.F+%\"hGF+)% \"dGFjrF+F+F+F-F+F+*,FdelF+%\"nGF+)%\"dGFaqF+)%\"cGFYF+)%\"eGF9F+F+*,F ezF+)%\"dGFevF+)%\"cGF]wF+)%\"gGF9F+)%\"eGF.F+F+*,F_glF+)%\"cGFjnF+)% \"hGF@F+)%\"eGF@F+)%\"dGFjrF+F+*,Fg_lF+)%\"dGFevF+)%\"cGFjnF+%\"aGF+)% \"gGF9F+F4*.\"%C5F+)%\"dGFgpF+)%\"cGFjpF+%\"hGF+)%\"eGF@F+)%\"gGF.F+F+ *.F_glF+)%\"cGFjnF+%\"hGF+)%\"eGF.F+)%\"gGF@F+)%\"dGFjrF+F4*.FcalF+)% \"dGFevF+)%\"cGF]wF+)%\"hGF.F+)%\"eGF@F+%\"gGF+F4*.FdelF+%\"nGF+)%\"dG FevF+)%\"cGFjnF+)%\"eGF@F+%\"hGF+F+*.FdelF+%\"nGF+)%\"dGFgpF+)%\"cGF]w F+)%\"eGF@F+%\"gGF+F4*.F_[lF+%\"nGF+)%\"cGFjrF+)%\"eGF.F+)%\"gGF.F+)% \"dGFjrF+F+*0\"&wX#F+)%\"dGFgpF+)%\"cGF]wF+%\"aGF+%\"eGF+%\"hGF+)%\"gG F.F+F4*0Fe]mF+)%\"dGFevF+)%\"cGFjnF+%\"aGF+%\"eGF+)%\"hGF.F+%\"gGF+F+* 0\"&/$)*F+)%\"dGFaqF+)%\"cGFYF+%\"aGF+)%\"eGF.F+%\"hGF+%\"gGF+F+*.\"&_ \"\\F+)%\"dGFRF+)%\"cGFjpF+%\"aGF+)%\"eGF.F+)%\"hGF.F+F4*.F]flF+)%\"cG FjrF+%\"aGF+%\"eGF+)%\"hGF@F+)%\"dGFjrF+F4*.F^blF+)%\"cGFjrF+%\"aGF+)% \"gGF@F+%\"hGF+)%\"dGFjrF+F+*.F]flF+)%\"dGFRF+)%\"cGFjpF+%\"aGF+%\"eGF +)%\"gGF@F+F+*.Fe_mF+)%\"dGF^tF+%\"aGF+)%\"eGF.F+)%\"gGF.F+)%\"cGF9F+F 4**F``lF+)%\"dG\"#=F+%\"aGF+)%\"eGF9F+F4*(\"$<\"F+)%\"cGFjqF+)%\"hGFjp F+F+**FdelF+)%\"cGFjrF+)F-F.F+)%\"dGF^tF+F+*,\"$(**F+)%\"cGFg\\lF+)%\" hGF9F+)%\"gGF.F+)%\"dGF.F+F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "discrim=0 [4]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1293 "disc rim(16*c^11*g^6*d^6+16*n*c^13*g^4*d^5+16*n*c^17*h^4*d-534*c^16*h^5*g*d -128*c^9*g^5*e*d^8+3024*c^11*h^4*e^2*d^6+1080*c^14*h^5*e*d^3-972*c^14* h^3*g^3*d^3+520*c^13*h^2*g^4*d^4-144*c^12*h*g^5*d^5-7488*c^10*h^3*e^2* g*d^7+6528*c^9*h^2*e^2*g^2*d^8-3816*c^13*h^4*e*g*d^4+5232*c^12*h^3*e*g ^2*d^5-3456*c^11*h^2*g^3*e*d^6+1088*c^10*h*g^4*e*d^7-256*n*c^11*e*g^3* d^7+1536*n*c^11*e^2*h^2*d^7+256*n*c^14*e*h^3*d^4+96*n*c^15*g^2*h^2*d^3 -64*n*c^14*g^3*h*d^4+768*n*c^12*e*h*g^2*d^6-64*n*c^16*g*h^3*d^2-768*n* c^13*e*h^2*g*d^5-512*c^12*a*h^4*d^6-131072*d^15*a*e^3*h*c^3+131072*d^1 6*a*e^3*g*c^2-3072*c^10*a*g^2*h^2*d^8+2048*c^11*a*g*h^3*d^7-3072*n*c^1 0*e^2*h*g*d^8+(1408*c^13*h^3*d^7-512*d^10*c^10*g^3+2048*d^12*c^8*e*g^2 -3200*c^12*g*h^2*d^8+4096*d^13*c^7*e^2*h-8192*d^11*c^9*e*h*g+6656*d^10 *c^10*e*h^2+2304*c^11*g^2*h*d^9)*_Z+4096*n*d^13*c^5*e^4+256*d^10*c^7*g ^4*e^2+2304*c^8*h^3*e^3*d^9-512*d^10*c^8*a*g^4+1024*d^11*c^6*h*e^3*g^2 -2304*c^8*h*e^2*g^3*d^9-3072*d^10*c^7*h^2*e^3*g+4096*n*d^10*c^8*e^3*h- 4096*n*d^11*c^7*e^3*g+1536*n*c^9*e^2*g^2*d^9-24576*d^11*c^7*a*e*h*g^2+ 24576*d^10*c^8*a*e*h^2*g+98304*d^13*c^5*a*e^2*h*g-49152*d^12*c^6*a*e^2 *h^2-8192*c^9*a*e*h^3*d^9+2048*c^9*a*g^3*h*d^9+8192*d^12*c^6*a*e*g^3-4 9152*d^14*a*e^2*g^2*c^4-131072*d^18*a*e^4+117*c^17*h^6+4096*c^9*_Z^2*d ^14+997*c^15*h^4*g^2*d^2,_Z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$** \"&Ob'\"\"\")%\"dG\"#9F&)%\"cG\"\"*F&,fp*,\"\"%F&%\"nGF&)%\"cG\"#8F&)% \"gGF/F&)%\"dG\"\"&F&!\"\"*,F/F&%\"nGF&)%\"cG\"#F&)%\"cG\"#5F&)%\"hGFinF&)% \"eGFSF&%\"gGF&)%\"dG\"\"(F&F9*.FKF&)%\"cGF,F&)%\"hGFSF&)%\"eGFSF&)%\" gGFSF&)%\"dG\"\")F&F&*.\"#[F&)%\"cGF3F&)%\"hGF/F&%\"eGF&%\"gGF&)%\"dGF /F&F9*.FfqF&)%\"cG\"#7F&)%\"hGFinF&%\"eGF&)%\"gGFSF&)%\"dGF8F&F&*.FEF& )%\"cGFNF&)%\"hGFSF&)%\"gGFinF&%\"eGF&)%\"dGFVF&F9*.\"#kF&%\"nGF&)%\"c GFNF&%\"eGF&)%\"gGFinF&)%\"dGFhpF&F&*.\"$%QF&%\"nGF&)%\"cGFNF&)%\"eGFS F&)%\"hGFSF&)%\"dGFhpF&F9*.FesF&%\"nGF&)%\"cGF)F&%\"eGF&)%\"hGFinF&)% \"dGF/F&F9*.\"#CF&%\"nGF&)%\"cG\"#:F&)%\"gGFSF&)%\"hGFSF&)%\"dGFinF&F9 *.FEF&%\"nGF&)%\"cGF)F&)%\"gGFinF&%\"hGF&)%\"dGF/F&F&*0F]pF&%\"nGF&)% \"cGFbrF&%\"eGF&%\"hGF&)%\"gGFSF&)%\"dGFVF&F9*.FEF&%\"nGF&)%\"cGFEF&% \"gGF&)%\"hGFinF&)%\"dGFSF&F&*0F]pF&%\"nGF&)%\"cGF3F&%\"eGF&)%\"hGFSF& %\"gGF&)%\"dGF8F&F&*,\"$G\"F&)%\"cGFbrF&%\"aGF&)%\"hGF/F&)%\"dGFVF&F&* .\"&oF$F&)%\"dGFguF&%\"aGF&)%\"eGFinF&%\"hGF&)%\"cGFinF&F&*.F^yF&)%\"d GFEF&%\"aGF&)%\"eGFinF&%\"gGF&)%\"cGFSF&F9*.\"$o(F&)%\"cGF`pF&%\"aGF&) %\"gGFSF&)%\"hGFSF&)%\"dGFdqF&F&*.\"$7&F&)%\"cGFNF&%\"aGF&%\"gGF&)%\"h GFinF&)%\"dGFhpF&F9*,\"$c#F&)%\"cGF8F&)%\"dGFbrF&)%\"eGF/F&)%\"hGFSF&F &*0FazF&%\"nGF&)%\"cGF`pF&)%\"eGFSF&%\"hGF&%\"gGF&)%\"dGFdqF&F&*,\"%C5 F&%\"nGF&)%\"dGF3F&)%\"cGF8F&)%\"eGF/F&F9*,Ff[lF&)%\"cGFdqF&)%\"hGFinF &)%\"eGFinF&)%\"dGF,F&F&*,FexF&)%\"dGF`pF&)%\"cGFdqF&%\"aGF&)%\"gGF/F& F&*.Ff[lF&)%\"dGF`pF&)%\"cGFhpF&)%\"hGFSF&)%\"eGFinF&%\"gGF&F9*.Fj\\lF &%\"nGF&)%\"dGF`pF&)%\"cGFdqF&)%\"eGFinF&%\"hGF&F9*.Fj\\lF&%\"nGF&)%\" dGFNF&)%\"cGFhpF&)%\"eGFinF&%\"gGF&F&*.F_tF&%\"nGF&)%\"cGF,F&)%\"eGFSF &)%\"gGFSF&)%\"dGF,F&F9*0\"%WhF&)%\"dGFNF&)%\"cGFhpF&%\"aGF&%\"eGF&%\" hGF&)%\"gGFSF&F&*0Fj`lF&)%\"dGF`pF&)%\"cGFdqF&%\"aGF&%\"eGF&)%\"hGFSF& %\"gGF&F9*0\"&wX#F&)%\"dGF3F&)%\"cGF8F&%\"aGF&)%\"eGFSF&%\"hGF&%\"gGF& F9*.\"&)G7F&)%\"dGFbrF&)%\"cGFVF&%\"aGF&)%\"eGFSF&)%\"hGFSF&F&*.\"%[?F &)%\"cGF,F&%\"aGF&%\"eGF&)%\"hGFinF&)%\"dGF,F&F&*.F\\[lF&)%\"cGF,F&%\" aGF&)%\"gGFinF&%\"hGF&)%\"dGF,F&F9*.FeclF&)%\"dGFbrF&)%\"cGFVF&%\"aGF& %\"eGF&)%\"gGFinF&F9*.FjblF&)%\"dGF)F&%\"aGF&)%\"eGFSF&)%\"gGFSF&)%\"c GF/F&F&**F^yF&)%\"dG\"#=F&%\"aGF&)%\"eGF/F&F&*&)%\"cGF>F&)%\"hGFVF&F&* ,FVF&)%\"cGFguF&)%\"hGF/F&)%\"gGFSF&)%\"dGFSF&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,\"&Ob'\"\"\")%\"dG\"#9F&)%\"cG\"\"*F&),(*(\"\"%F&%\"eGF&)%\" dG\"\"$F&F&*(%\"gGF&)%\"cG\"\"#F&%\"dGF&!\"\"*&%\"hGF&)%\"cGF4F&F&F0F& ,(*(\"$G\"F&)%\"dG\"\"'F&%\"aGF&F&**F0F&%\"dGF&)%\"cG\"\"&F&%\"nGF&F;* &)%\"cGFKF&)%\"hGF9F&F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Ag ain, the shorter factor is the same as in case B [7]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "solve(128*d^6*a-4*d*c^5*n+c^5*h^2,n );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"%!\"\",&*(\"$G\"\"\"\")% \"dG\"\"'F*%\"aGF*F**&)%\"cG\"\"&F*)%\"hG\"\"#F*F*F*%\"dGF&%\"cG!\"&F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A7:=eval(A6,n=%);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#A7G,<*&%\"aG\"\"\")%\"xG\"\"&F(F(*, \"\"%!\"\"%\"cG\"\"#%\"dGF.%\"xGF0%\"yGF(F(*(%\"cGF(%\"xGF()%\"yGF0F(F (*&%\"dGF()%\"yG\"\"$F(F(*(%\"eGF()%\"xGF=F(%\"yGF(F(*.F-F.,(**F0F(%\" gGF()%\"cGF0F(%\"dGF(F(*(\"\")F(%\"eGF()%\"dGF=F(F(*&%\"hGF()%\"cGF=F( F.F(%\"cGF.%\"dG!\"#%\"xGF0%\"yGF0F(*(%\"gGF(%\"xGF()%\"yGF=F(F(*&%\"h GF()%\"yGF-F(F(*.\"#;F.,8**\"#KF()%\"dG\"\"'F(%\"cGF(%\"aGF(F(*(F-F.% \"cGF(,,*,F\\oF()%\"dGF=F()%\"cGF0F(%\"hGF(%\"eGF(F(*(\"%!G\"F()%\"dGF boF(%\"aGF(F(*&)%\"cGF+F()%\"hGF0F(F(*,F-F(%\"dGF()%\"cGF-F(%\"gGF(%\" hGF(F.**F-F()%\"dGF0F()%\"cGF=F()%\"gGF0F(F(F(F.*(F0F.%\"cGF(,(*(\"$?$ F()%\"dGFboF(%\"aGF(F(*&)%\"cGF+F()%\"hGF0F(F.*,F0F(%\"dGF()%\"cGF-F(% \"gGF(%\"hGF(F(F(F(*()%\"dGF0F()%\"cGF-F()%\"gGF0F(F(*(F-F.%\"cGF(,&*( \"$G\"F()%\"dGFboF(%\"aGF(F(*&)%\"cGF+F()%\"hGF0F(F(F(F.*&%\"cGF(,(*,F KF()%\"dGF-F(%\"cGF(%\"gGF(%\"eGF(F(*,F-F()%\"dGF=F()%\"cGF0F(%\"hGF(% \"eGF(F.*(FjqF()%\"dGFboF(%\"aGF(F(F(F(*,FKF()%\"dGF-F()%\"cGF0F(%\"gG F(%\"eGF(F.*(F\\oF()%\"dGFboF()%\"eGF0F(F(*,F0F(%\"dGF()%\"cGF+F(%\"gG F(%\"hGF(F.*,FKF()%\"dGF=F()%\"cGF=F(%\"hGF(%\"eGF(F(*&)%\"cGFboF()%\" hGF0F(F(F(%\"dG!\"&%\"cGFU%\"xGF-%\"yGF(F(*.FKF.,(*,FKF()%\"dGF-F(%\"c GF(%\"gGF(%\"eGF(F(*,F-F()%\"dGF=F()%\"cGF0F(%\"hGF(%\"eGF(F.*(FjqF()% \"dGFboF(%\"aGF(F(F(%\"dG!\"%%\"cGFU%\"xGF=%\"yGF0F(*.F\\oF.,,*,F\\oF( )%\"dGF=F()%\"cGF0F(%\"hGF(%\"eGF(F(*(F`pF()%\"dGFboF(%\"aGF(F(*&)%\"c GF+F()%\"hGF0F(F(*,F-F(%\"dGF()%\"cGF-F(%\"gGF(%\"hGF(F.**F-F()%\"dGF0 F()%\"cGF=F()%\"gGF0F(F(F(%\"cG!\"$%\"dGF\\[l%\"xGF0%\"yGF=F(*.F-F.,(* (FjqF()%\"dGFboF(%\"aGF(F(*&)%\"cGF+F()%\"hGF0F(F.*,F0F(%\"dGF()%\"cGF -F(%\"gGF(%\"hGF(F(F(%\"dGFU%\"cGFhx%\"xGF(%\"yGF-F(*,F-F.,&*(FdsF()% \"dGFboF(%\"aGF(F(*&)%\"cGF+F()%\"hGF0F(F(F(%\"dGF.%\"cGF`w%\"yGF+F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "reevaluate polynomial [3]" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A7,x=0,y,0);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#<%\"\"!,,*.\"(w&[5!\"\"%\"aG\"\"&,L*&) %\"cG\"#:\"\"\")%\"hGF*F0F0*()%\"cG\"#5F0)%\"gGF*F0)%\"dGF*F0F(*.\"$![ F0)%\"dG\"\")F0)%\"cG\"\"(F0%\"hGF0)%\"eG\"\"#F0)%\"gGFFF0F0*,\"$g\"F0 )%\"dG\"\"*F0)%\"cG\"\"'F0)%\"gG\"\"$F0)%\"eGFFF0F(*,\"#?F0)%\"cGF?F0) %\"gG\"\"%F0%\"eGF0)%\"dGFBF0F0*,FJF0)%\"cGFMF0)%\"hGFSF0)%\"eGFFF0)% \"dGFPF0F0*,FWF0)%\"cG\"#7F0)%\"hGFfnF0%\"eGF0)%\"dGFSF0F0*,F6F0)%\"cG FfoF0)%\"hGFFF0)%\"gGFSF0)%\"dGFSF0F(*,F6F0)%\"cG\"#8F0)%\"hGFSF0)%\"g GFFF0)%\"dGFFF0F0*.F " 0 "" {MPLTEXT 1 0 400 "factor(c^15*h^5-c^10*g^5*d^5+480*d^8*c^7*h*e^2*g^2-1 60*d^9*c^6*g^3*e^2+20*c^8*g^4*e*d^7+160*c^9*h^3*e^2*d^6+20*c^12*h^4*e* d^3-10*c^12*h^2*g^3*d^3+10*c^13*h^3*g^2*d^2-480*c^8*h^2*e^2*g*d^7-80*c ^9*h*e*g^3*d^6-1280*d^10*g*e^3*h*c^5+5*c^11*h*g^4*d^4+640*d^9*c^6*h^2* e^3-5*c^14*h^4*g*d+1024*d^15*e^5+120*c^10*h^2*e*g^2*d^5-80*c^11*h^3*e* g*d^4+640*d^11*e^3*g^2*c^4+1280*d^12*e^4*h*c^3-1280*d^13*e^4*g*c^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),(*(\"\"%\"\"\"%\"eGF()%\"dG\"\"$ F(F(*(%\"gGF()%\"cG\"\"#F(%\"dGF(!\"\"*&%\"hGF()%\"cGF,F(F(\"\"&F(" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "This is the same factor as in cas e B so all cases are exhausted." }}}}{MARK "58 0 0" 64 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }