{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 }} {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 65 "A:=y^2+2*x^2*y+x^4+a*x^3*y+b *x^2*y^2+c*x*y^3+d*y^4+e*x*y^2+f*y^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,4*$)%\"yG\"\"#\"\"\"F**(F)F*)%\"xGF)F*F(F*F**$)F-\"\"%F*F**( %\"aGF*)F-\"\"$F*F(F*F**(%\"bGF*F,F*F'F*F**(%\"cGF*F-F*)F(F4F*F**&%\"d GF*)F(F0F*F**(%\"eGF*F-F*F'F*F**&%\"fGF*F9F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "First we define the polynomial based on the Newton P olygon. [1]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "puiseux(A,x= 0,y,0);" }{TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'RootOf G6#,(*&%\"dG\"\"\")%#_ZG\"\"#F*F**&%\"fGF*F,F*F*F*F*,&*&),&%\"aG!\"\"% \"eGF*\"\"$F*)*&%\"xGF*,&F4F*F6F5F5#\"\"&F-F*F5*(F3F-F:F-F;!\"#F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "The Puiseux indicates that the je ts split at 5/2, so set the nominator of the 5/2 term equal to zero. T his forces a higher splitting because then the coefficients will both \+ be zero [2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A1:=eval(A, a=e);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,4*$)%\" yG\"\"#\"\"\"F**(F)F*)%\"xGF)F*F(F*F**$)F-\"\"%F*F**(%\"eGF*)F-\"\"$F* F(F*F**(%\"bGF*F,F*F'F*F**(%\"cGF*F-F*)F(F4F*F**&%\"dGF*)F(F0F*F**(F2F *F-F*F'F*F**&%\"fGF*F9F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "We evaluate the polynomial where the numerator of the 5/2 term in the Pu iseux jet is equal to zero [3]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A1,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'R ootOfG6#,(*&%\"dG\"\"\")%#_ZG\"\"#F*F**&%\"fGF*F,F*F*F*F*,&*$)%\"xGF-F *!\"\"*&)F3\"\"$F*-F%6#,**&%\"eGF*F,F*F4*$F+F*F*%\"bGF*F/F4F*F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Here the jets split at the whole \+ power of 3, so, for the coefficients to be equal, we set the discrimin ant of the RootOf( ) expression equal to zero [4]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A2:=eval(A1,b=f+e^2/4);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#A2G,4*$)%\"yG\"\"#\"\"\"F**(F)F*)%\"xGF)F*F(F*F**$ )F-\"\"%F*F**(%\"eGF*)F-\"\"$F*F(F*F**(,&%\"fGF**&F0!\"\"F2F)F*F*F,F*F 'F*F**(%\"cGF*F-F*)F(F4F*F**&%\"dGF*)F(F0F*F**(F2F*F-F*F'F*F**&F7F*F " 0 "" {MPLTEXT 1 0 20 "puiseux(A2,x=0,y,0) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$-%'RootOfG6#,(*&%\"dG\"\"\")%#_ ZG\"\"#F*F**&%\"fGF*F,F*F*F*F*,(*(\"#;!\"\",&*&F/F*%\"eGF*F**&F-F*%\"c GF*F3\"\"%*&%\"xGF*,&*(F-F3F/F*F6F*F3F8F*F3#\"\"(F-F**,F2F3F6F*F4\"\"$ F;FAF " 0 "" {MPLTEXT 1 0 21 "A3:=eval(A2,c=f*e/2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G,4*$)%\"yG\"\"#\"\"\"F**(F)F*)% \"xGF)F*F(F*F**$)F-\"\"%F*F**(%\"eGF*)F-\"\"$F*F(F*F**(,&%\"fGF**&F0! \"\"F2F)F*F*F,F*F'F*F**,F)F9F7F*F2F*F-F*F(F4F**&%\"dGF*)F(F0F*F**(F2F* F-F*F'F*F**&F7F*)F(F4F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Reev aluate the polynomial [3]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A3,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,(*$)%\" xG\"\"#\"\"\"!\"\"*(F(F*%\"eGF)F'\"\"$F)*&)F'\"\"%F)-%'RootOfG6#,,*&\" #;F))%#_ZGF(F)F)*&,&*&\"\")F))F,F(F)F)*&F6F)%\"fGF)F)F)F8F)F)*$)F,F0F) F)*(F0F)F?F)F=F)F)*&F6F)%\"dGF)F)F)F)-F26#,(*&FDF)F7F)F)*&F?F)F8F)F)F) F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Set the disicriminant of th e RootOf( ) expression in the x^4 term equal to zero [4]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "discrim(16*_Z^2+(8*e^2+16*f)*_Z+e^4 +4*f*e^2+16*d,_Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"%C5\"\"\"% \"dGF&!\"\"*&\"$c#F&)%\"fG\"\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A4:=eval(A3,d=-f^2/4);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#>%#A4G,4*$)%\"yG\"\"#\"\"\"F**(F)F* )%\"xGF)F*F(F*F**$)F-\"\"%F*F**(%\"eGF*)F-\"\"$F*F(F*F**(,&%\"fGF**&F0 !\"\"F2F)F*F*F,F*F'F*F**,F)F9F7F*F2F*F-F*F(F4F**(F0F9F7F)F(F0F9*(F2F*F -F*F'F*F**&F7F*)F(F4F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Note that A4 is now a reducible curve [5]. We cannot factor using Maple, b ut it can be shown that it is reducible." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "puiseux(A4,x=0,y,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$,(*$)%\"xG\"\"#\"\"\"!\"\"*(F(F*%\"eGF)F'\"\"$F)*&)F'\"\"%F)-%'Ro otOfG6#,,*&\"#;F))%#_ZGF(F)F)*&,&*&\"\")F))F,F(F)F)*&F6F)%\"fGF)F)F)F8 F)F)*$)F,F0F)F)*(F0F)F?F)F=F)F)*&F0F))F?F(F)F*F)F)-F26#,(*&FDF)F7F)F)* (F0F)F?F)F8F)F*F0F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Note that \+ this Puiseux expansion does not split at a higher power." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "discrim(16*_Z^2+(8*e^2+16*f)*_Z+e^4 +4*f*e^2-4*f^2,_Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"$7&\"\"\" )%\"fG\"\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A5:=eval (A4,f=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G,.*$)%\"yG\"\"#\"\" \"F**(F)F*)%\"xGF)F*F(F*F**$)F-\"\"%F*F**(%\"eGF*)F-\"\"$F*F(F*F**&#F* F0F**(F'F*F,F*)F2F)F*F*F**(F2F*F-F*F'F*F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "factor(A5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*& \"\"%!\"\",(*&\"\"#\"\"\")%\"xGF)F*F**(%\"yGF*F,F*%\"eGF*F**&F)F*F.F*F *F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "Maple shows that A5 fac tors [5]. This is normally how we know we are done because the curve \+ is reducible." }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }