{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 } {CSTYLE "" 20 256 "" 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 1 12 0 0 0 0 0 2 2 0 0 0 0 0 0 1 }{CSTYLE "" 20 281 "" 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 20 282 "" 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 20 283 "" 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 Out put" -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 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT -1 49 "Chapter 20\n\nA Bird's- Eye View of Groebner Bases \n" }}{PARA 0 "" 0 "" {TEXT 280 31 "\251 Co pyright 2003 by Andr\351 Heck." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 0 "" 0 "" {TEXT 257 2 "1." }{TEXT -1 1 " " }{TEXT 256 4 "Let " }{XPPEDIT 18 0 "f = x^2*y^2-w^2+w,f[1] = x-y^2*w,f[2] = y-z*w,f [3] = z-w^3;" "6&/%\"fG,(*&%\"xG\"\"#%\"yGF(\"\"\"*$%\"wGF(!\"\"F,F*/& F$6#F*,&F'F**&F)F(F,F*F-/&F$6#F(,&F)F**&%\"zGF*F,F*F-/&F$6#\"\"$,&F8F* *$F,F " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebner);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#77%*MulMatrixG%)SetBasisG%%fglmG%'gbasisG%'gsol veG%+hilbertdimG%,hilbertpolyG%.hilbertseriesG%-inter_reduceG%*is_fini teG%,is_solvableG%*leadcoeffG%(leadmonG%)leadtermG%(normalfG%/pretend_ gbasisG%'reduceG%&spolyG%*termorderG%*testorderG%)univpolyG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := x^2*y^2-w^2+w;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*&)%\"xG\"\"#\"\"\")%\"yGF)F*F **$)%\"wGF)F*!\"\"F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f 1 := x-y^2*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G,&%\"xG\"\"\"*& )%\"yG\"\"#F'%\"wGF'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f2 := y-z*w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G,&%\"yG\"\" \"*&%\"zGF'%\"wGF'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " f3 := z-w^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f3G,&%\"zG\"\"\"*$) %\"wG\"\"$F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f4 := \+ w^3-w;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f4G,&*$)%\"wG\"\"$\"\"\"F *F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "gb1 := gbasis([ f1,f2,f3,f4], plex(x,y,z,w));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gb 1G7&,&*$)%\"wG\"\"$\"\"\"F+F)!\"\",&%\"zGF+F)F,,&%\"yGF+*$)F)\"\"#F+F, ,&%\"xGF+F)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "normalf(f, gb1, plex(x,y,z,w));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"wG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "gb2 := gbasis([f1,f2,f3,f4], plex(w,z,y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$gb2G7&,&*$)%\"x G\"\"$\"\"\"F+F)!\"\",&%\"yGF+*$)F)\"\"#F+F,,&F)F,%\"zGF+,&%\"wGF+F)F, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "normalf(f, gb2, plex(w, z,y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"xG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 258 2 "2. " }{TEXT -1 42 " Find the common zeros of the polynomials " }{XPPEDIT 18 0 "x*y*z-w;" "6#,&*(%\"xG\"\"\"%\"yGF&%\"zGF&F&%\"wG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y*z*w-x;" "6#,&*(%\"yG\"\"\"%\"zGF&%\"wGF&F &%\"xG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "z*w*x-y;" "6#,&*(%\"zG \"\"\"%\"wGF&%\"xGF&F&%\"yG!\"\"" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "w*x*y-z;" "6#,&*(%\"wG\"\"\"%\"xGF&%\"yGF&F&%\"zG!\"\"" }{TEXT -1 34 " using the Groebner basis method.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(Groebner):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "gb := gbasis([x*y*z-w, y*z*w-x, z*w*x-y, w*x*y-z], \+ plex(x,y,z,w));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gbG7(,&%\"wG!\" \"*$)F'\"\"&\"\"\"F,,&%\"zGF(*&)F'\"\"%F,F.F,F,,&*$)F.\"\"#F,F,*$)F'F5 F,F(,&%\"yGF(*&F0F,F9F,F,,&*$)F9F5F,F,F6F(,&*(F9F,F.F,F'F,F(%\"xGF," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "factor(%[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"wG\"\"\",&F$F%F%!\"\"F%,&F$F%F%F%F%,&*$)F$ \"\"#F%F%F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "So: " } {XPPEDIT 18 0 "w = 0,1,-1,I,-I;" "6'/%\"wG\"\"!\"\"\",$F&!\"\"%\"IG,$F )F(" }{TEXT -1 16 ". Note that for " }{XPPEDIT 18 0 "w <> 0;" "6#0%\"w G\"\"!" }{TEXT -1 7 " holds " }{XPPEDIT 18 0 "w^4 = 1;" "6#/*$%\"wG\" \"%\"\"\"" }{TEXT -1 20 ". For each value of " }{TEXT 284 1 "w" } {TEXT -1 71 " , the other variables can easily be solved via the other polynomials: " }{XPPEDIT 18 0 "y=w" "6#/%\"yG%\"wG" }{TEXT -1 4 " or \+ " }{XPPEDIT 18 0 "y=-w" "6#/%\"yG,$%\"wG!\"\"" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "z = w;" "6#/%\"zG%\"wG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "z = -w;" "6#/%\"zG,$%\"wG!\"\"" }{TEXT -1 31 ". For each choice \+ the value of " }{TEXT 285 1 "x" }{TEXT -1 15 " is determined." }} {PARA 0 "" 0 "" {TEXT -1 63 "Note that we must do siome work in Maple \+ to find all solutions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "s olve(\{op(gb)\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6,<&/%\"zG\"\"!/%\" wGF&/%\"yGF&/%\"xGF&<&/F(\"\"\"/F*F//F%F//F,F/<&F.F0/F%!\"\"/F,F5<&F./ F*F5F1F6<&F.F8F4F2<&/F%-%'RootOfG6#,&*$)%#_ZG\"\"#F/F/F/F//F(F " 0 "" {MPLTEXT 1 0 46 "candidates := map(allvalues,[%], independen t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+candidatesG7;<&/%\"zG\"\"!/% \"wGF)/%\"yGF)/%\"xGF)<&/F+\"\"\"/F-F2/F(F2/F/F2<&F1F3/F(!\"\"/F/F8<&F 1/F-F8F4F9<&F1F;F7F5<&/F+^#F2/F-F?/F(F?/F/^#F8<&F>F@FB/F(FC<&F@FAFB/F+ FC<&F@FBFGFE<&F>FAFB/F-FC<&F>FBFJFE<&FAFBFJFG<&FBFJFGFE<&F>F@FA/F/F?<& F>F@FOFE<&F@FAFOFG<&F@FOFGFE<&F>FAFOFJ<&F>FOFJFE<&FAFOFJFG<&FOFJFGFE<& F3F4/F+F8F9<&F3F7FXF5<&F;F4FXF5<&F;F7FXF9" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "nops(candidates);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sols := select(x- >eval(gb,x)=[0$6], candidates);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%% solsG73<&/%\"zG\"\"!/%\"wGF)/%\"yGF)/%\"xGF)<&/F+\"\"\"/F-F2/F(F2/F/F2 <&F1F3/F(!\"\"/F/F8<&F1/F-F8F4F9<&F1F;F7F5<&/F+^#F2/F-F?/F(F?/F/^#F8<& F@FB/F+FC/F(FC<&F>FB/F-FCFF<&FAFBFHFE<&F>F@/F/F?FF<&F@FAFKFE<&F>FAFKFH <&FKFHFEFF<&F3F4/F+F8F9<&F3F7FPF5<&F;F4FPF5<&F;F7FPF9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nops(sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "is_s olvable(gb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "is_finite(gb);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "h ilbertseries(gb, tdeg(x,y,z,w), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,,*$)%\"tG\"\"%\"\"\"F(*&F'F()F&\"\"$F(F(*&\"\"(F()F&\"\"#F(F(*&F'F(F &F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eval(%,t=1); # number of solutions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 259 2 "3." }{TEXT -1 102 " Consider the chemical reaction mech anism whose steady state is described by the polynomial equations\n" } {XPPEDIT 18 0 "0 = -beta*c[1]^2-gamma*c[1]*c[2]+alpha*c[1]+epsilon*c[3 ];" "6#/\"\"!,**&%%betaG\"\"\"*$&%\"cG6#F(\"\"#F(!\"\"*(%&gammaGF(&F+6 #F(F(&F+6#F-F(F.*&%&alphaGF(&F+6#F(F(F(*&%(epsilonGF(&F+6#\"\"$F(F(" } {TEXT -1 1 "\n" }{XPPEDIT 18 0 "0 = -gamma*c[1]*c[2]-eta*c[2]^2+epsilo n*c[3]+2*theta*c[3];" "6#/\"\"!,**(%&gammaG\"\"\"&%\"cG6#F(F(&F*6#\"\" #F(!\"\"*&%$etaGF(*$&F*6#F.F.F(F/*&%(epsilonGF(&F*6#\"\"$F(F(*(F.F(%&t hetaGF(&F*6#F9F(F(" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "0 = gamma*c[1]*c[ 2]+eta*c[2]^2-epsilon*c[3]-theta*c[3];" "6#/\"\"!,**(%&gammaG\"\"\"&% \"cG6#F(F(&F*6#\"\"#F(F(*&%$etaGF(*$&F*6#F.F.F(F(*&%(epsilonGF(&F*6#\" \"$F(!\"\"*&%&thetaGF(&F*6#F8F(F9" }{TEXT -1 126 "\n\nA small change i n the reaction mechanism has a big effect: consider the steady states \+ described by the polynomial equations\n" }{XPPEDIT 18 0 "0 = -beta*c[1 ]^2-gamma*c[1]*c[2]+alpha*c[1]+epsilon*c[3];" "6#/\"\"!,**&%%betaG\"\" \"*$&%\"cG6#F(\"\"#F(!\"\"*(%&gammaGF(&F+6#F(F(&F+6#F-F(F.*&%&alphaGF( &F+6#F(F(F(*&%(epsilonGF(&F+6#\"\"$F(F(" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "0 = -gamma*c[1]*c[2]-eta*c[2]+epsilon*c[3]+theta*c[3];" "6#/\"\" !,**(%&gammaG\"\"\"&%\"cG6#F(F(&F*6#\"\"#F(!\"\"*&%$etaGF(&F*6#F.F(F/* &%(epsilonGF(&F*6#\"\"$F(F(*&%&thetaGF(&F*6#F8F(F(" }{TEXT -1 1 "\n" } {XPPEDIT 18 0 "0 = gamma*c[1]*c[2]+eta*c[2]-epsilon*c[3]-theta*c[3]" " 6#/\"\"!,**(%&gammaG\"\"\"&%\"cG6#F(F(&F*6#\"\"#F(F(*&%$etaGF(&F*6#F.F (F(*&%(epsilonGF(&F*6#\"\"$F(!\"\"*&%&thetaGF(&F*6#F7F(F8" }{TEXT -1 98 "\nShow that this reaction mechanism has a one-dimensional solution space of positive steady states\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(Groebner):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "alias(seq(c[k]=c||k, k=1..3)):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 147 "polys := [-beta*c1^2-gamma*c1*c2+alpha*c1+e psilon*c3,\n -gamma*c1*c2-2*eta*c2^2+epsilon*c3+2*theta*c3,\n gamma* c1*c2+eta*c2^2-epsilon*c3-theta*c3];" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&polysG7%,**&%%betaG\"\"\")&%\"cG6#F)\"\"#F)!\"\"*(%&gammaGF)F+F)& F,6#F.F)F/*&%&alphaGF)F+F)F)*&%(epsilonGF)&F,6#\"\"$F)F),*F0F/*(F.F)%$ etaGF))F2F.F)F/F6F)*(F.F)%&thetaGF)F8F)F),*F0F)*&F=F)F>F)F)F6F/*&F@F)F 8F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "reactionConstants \+ := alpha, beta, gamma, epsilon, eta, theta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2reactionConstantsG6(%&alphaG%%betaG%&gammaG%(epsilon G%$etaG%&thetaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "concentr ations := c||(1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/concentrati onsG6%&%\"cG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "vars := reactionConstants, concentrations:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gbasis(polys, plex(vars));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*&%$etaG\"\"\")&%\"cG6#\"\"#F,F' F'*&%&thetaGF'&F*6#\"\"$F'!\"\",&*(%&gammaGF'&F*6#F'F'F)F'F2*&%(epsilo nGF'F/F'F',&*&%&alphaGF'F6F'F'*&%%betaGF')F6F,F'F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "solve(\{op(%)\}, \{c1,c2,c3\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&<%/&%\"cG6#\"\"#\"\"!/&F&6#\"\"$F)/&F&6#\"\" \"F)F#<%F$F*/F/*&%&alphaGF1%%betaG!\"\"<%F3/F%*.%&gammaGF1F5F1%&thetaG F1%(epsilonGF7%$etaGF7F6F7/F+*.F>F7F;F(F5F(F \+ " 0 "" {MPLTEXT 1 0 139 "polys := [-beta*c1^2-gamma*c1*c2+alpha*c1+eps ilon*c3,\n -gamma*c1*c2-eta*c2+epsilon*c3+theta*c3,\n gamma*c1*c2+et a*c2-epsilon*c3-theta*c3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polys G7%,**&%%betaG\"\"\")&%\"cG6#F)\"\"#F)!\"\"*(%&gammaGF)F+F)&F,6#F.F)F/ *&%&alphaGF)F+F)F)*&%(epsilonGF)&F,6#\"\"$F)F),*F0F/*&%$etaGF)F2F)F/F6 F)*&%&thetaGF)F8F)F),*F0F)FF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "reactionConstants := alpha, beta, gamma, epsilon, eta , theta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2reactionConstantsG6(%&a lphaG%%betaG%&gammaG%(epsilonG%$etaG%&thetaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "concentrations := c||(1..3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%/concentrationsG6%&%\"cG6#\"\"\"&F'6#\"\"#&F'6#\"\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "vars := reactionConsta nts, concentrations:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gba sis(polys, plex(vars));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,**(%&gam maG\"\"\"&%\"cG6#F'F'&F)6#\"\"#F'!\"\"*&%$etaGF'F+F'F.*&%(epsilonGF'&F )6#\"\"$F'F'*&%&thetaGF'F3F'F',**&%&alphaGF'F(F'F'*&%%betaGF')F(F-F'F. F/F'F6F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(\{op(%)\} , \{c2,c3\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"cG6#\"\"$,$*(& F&6#\"\"\"F-,**(%&alphaGF-%&gammaGF-F+F-!\"\"*(%%betaGF-)F+\"\"#F-F1F- F-*&%$etaGF-F0F-F2*(F8F-F4F-F+F-F-F-,&*(F1F-F+F-%&thetaGF-F-*&F8F-%(ep silonGF-F2F2F2/&F&6#F6,$*(F+F-,**&F0F-F>F-F2*(F>F-F4F-F+F-F-*&F0F-F " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(Gr oebner):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sys1 := [x^2-2* x+5, x*y^2+y*z^3, 3*y^2-8*z^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% sys1G7%,(*$)%\"xG\"\"#\"\"\"F+*&F*F+F)F+!\"\"\"\"&F+,&*&F)F+)%\"yGF*F+ F+*&F2F+)%\"zG\"\"$F+F+,&*&F6F+F1F+F+*&\"\")F+F4F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "is_solvable(sys1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "i s_finite(sys1, [x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "hilbertseries(sys1, tdeg(x ,y,z), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"tG\"\"&\"\"\"F(* &\"\"$F()F&\"\"%F(F(*&F'F()F&F*F(F(*&F'F()F&\"\"#F(F(*&F*F(F&F(F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "eval(%, t=1); # number \+ of solutions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#=" }}}{PARA 0 "" 0 "" {TEXT -1 22 "We verify this number:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gsys1 := gbasis(sys1, plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&gsys1G7',(*&\"$W\"\"\"\")%\"zG\"\"'F)F)*&\"\"*F)) F+F.F)F)*&\"%+;F))F+\"\"$F)F),(*&\"#SF)F2F)F)*(F,F)%\"yGF)F2F)F)*&F3F) F*F)F),&*&F3F))F8\"\"#F)F)*&\"\")F)F2F)!\"\",(*(\"#;F)%\"xGF)F2F)F)*&F 3F)F*F)F@*&F6F)F2F)F@,(*$)FDF=F)F)*&F=F)FDF)F@\"\"&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "algsubs(z^3=Z, gsys1[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&\"%+;\"\"\"%\"ZGF&F&*&\"\"*F&)F'\"\"$F&F &*&\"$W\"F&)F'\"\"#F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 " solve(%, Z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"!^$!\")#\"#K\"\"$^ $F%#!#KF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "map2(algsubs, \+ z^3=Z, gsys1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7',(*&\"%+;\"\"\"%\" ZGF'F'*&\"\"*F')F(\"\"$F'F'*&\"$W\"F')F(\"\"#F'F',(*(\"\"'F'%\"yGF'F(F 'F'*&F,F'F/F'F'*&\"#SF'F(F'F',&*&F,F')F4F0F'F'*&\"\")F'F(F'!\"\",(*&F7 F'F(F'F=*&F,F'F/F'F=*(\"#;F'%\"xGF'F(F'F',(*$)FCF0F'F'*&F0F'FCF'F=\"\" &F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "simplify(subs(Z=-8+3 2/3*I, %));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"!,(*&\"#[\"\"\"% \"yGF(!\"\"*&^#\"#kF(F)F(F(^$#!%39\"\"$#!$c#F1F(,&*&F1F()F)\"\"#F(F(^$ F-F2F(,(^$#\"%39F1#\"$c#F1F(*&\"$G\"F(%\"xGF(F**&^##\"$7&F1F(FAF(F(,(* $)FAF7F(F(*&F7F(FAF(F*\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(\{op(%)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"yG^$# !\")\"\"$#!#;F)/%\"xG^$\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplify(subs(Z=-8-32/3*I, %%%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7'\"\"!,(*&\"#[\"\"\"%\"yGF(!\"\"*&^#!#kF(F)F(F(^$#!% 39\"\"$#\"$c#F1F(,&*&F1F()F)\"\"#F(F(^$\"#kF2F(,(^$#\"%39F1#!$c#F1F(*& \"$G\"F(%\"xGF(F**&^##!$7&F1F(FBF(F(,(*$)FBF7F(F(*&F7F(FBF(F*\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(\{op(%)\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"xG^$\"\"\"!\"#/%\"yG^$#!\")\"\"$ #\"#;F." }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 262 4 "(ii)" }{TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(Groebner) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sys2 := [x^2*z^2+x^3, \+ x*z^4+2*x^2*z^2+x^3, y^2*z-2*y*z^2+z^3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sys2G7%,&*&)%\"xG\"\"#\"\"\")%\"zGF*F+F+*$)F)\"\"$F+F+,(*&F)F +)F-\"\"%F+F+*(F*F+F(F+F,F+F+F.F+,(*&)%\"yGF*F+F-F+F+*(F*F+F9F+F,F+!\" \"*$)F-F0F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "is_solvabl e(sys2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "is_finite(sys2, [x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 47 "We compu te the dimension of the solution space." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "hilbertdim(sys2, plex(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 263 2 "5." }{TEXT -1 102 " The metho d of Lagrange multipliers tells that if a critical point (maximum or m inimum) of a function " }{XPPEDIT 18 0 "f(x[1],`...`,x[n]);" "6#-%\"fG 6%&%\"xG6#\"\"\"%$...G&F'6#%\"nG" }{TEXT -1 26 " subject to the condit ion " }{XPPEDIT 18 0 "g(x[1],`...`,x[n]) = 0;" "6#/-%\"gG6%&%\"xG6#\" \"\"%$...G&F(6#%\"nG\"\"!" }{TEXT -1 41 " exists, it is attained at a \+ point where " }{XPPEDIT 18 0 "diff(f+lambda*g,x[i]) = 0;" "6#/-%%diffG 6$,&%\"fG\"\"\"*&%'lambdaGF)%\"gGF)F)&%\"xG6#%\"iG\"\"!" }{TEXT -1 10 " for some " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 88 ". Us e this method to find the radius of the largest sphere inscribable in \+ the ellipsoid " }{XPPEDIT 18 0 "x^2+2*y^2+3*z^2 = 4;" "6#/,(*$%\"xG\" \"#\"\"\"*&F'F(*$%\"yGF'F(F(*&\"\"$F(*$%\"zGF'F(F(\"\"%" }{TEXT -1 68 ". What is the radius of the smallest sper containing the ellipsoid?\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(Groebner) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "sys := [x^2+2*y^2+3*z ^2-4,\ndiff(x^2+y^2+z^2+lambda*(x^2+2*y^2+3*z^2-4), x),\ndiff(x^2+y^2+ z^2+lambda*(x^2+2*y^2+3*z^2-4), y),\ndiff(x^2+y^2+z^2+lambda*(x^2+2*y^ 2+3*z^2-4), z)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sysG7&,**$)%\"x G\"\"#\"\"\"F+*&F*F+)%\"yGF*F+F+*&\"\"$F+)%\"zGF*F+F+\"\"%!\"\",&*&F*F +F)F+F+*(F*F+%'lambdaGF+F)F+F+,&*&F*F+F.F+F+*(F3F+F8F+F.F+F+,&*&F*F+F2 F+F+*(\"\"'F+F8F+F2F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " gb := gbasis(sys, plex(x,y,z,lambda));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#gbG7,,**&\"\"'\"\"\")%'lambdaG\"\"$F)F)*&\"#6F))F+\"\"#F)F)*& F(F)F+F)F)F)F),&%\"zGF)*(F,F)F+F)F3F)F),**&\"#7F)F/F)!\"\"*&\"#=F)F+F) F8*$)F3F0F)F)F(F8,&%\"yGF)*(F0F)F+F)F>F)F)*&F3F)F>F),**$)F>F0F)F)*&\"# CF)F/F)F)*&\"#KF)F+F)F)\"\")F),&%\"xGF)*&F+F)FJF)F)*&F3F)FJF)*&F>F)FJF ),**$)FJF0F)F)*&F7F)F/F)F8*&\"#5F)F+F)F8F0F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "factor(gb);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 ,*(,&\"\"\"F&*&\"\"#F&%'lambdaGF&F&F&,&F&F&*&\"\"$F&F)F&F&F&,&F&F&F)F& F&*&%\"zGF&F*F&,**&\"#7F&)F)F(F&!\"\"*&\"#=F&F)F&F4*$)F/F(F&F&\"\"'F4* &%\"yGF&F%F&*&F/F&F;F&,**$)F;F(F&F&*&\"#CF&F3F&F&*&\"#KF&F)F&F&\"\")F& *&%\"xGF&F-F&*&F/F&FFF&*&F;F&FFF&,**$)FFF(F&F&*&F2F&F3F&F4*&\"#5F&F)F& F4F(F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(\{op(%)\}); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6&<&/%\"yG\"\"!/%\"zGF&/%'lambdaG!\" \"/%\"xG\"\"#<&F$F'F)/F-!\"#<&/F-F&F'/F*#F+F./F%-%'RootOfG6$,&*$)%#_ZG F.\"\"\"F>F.F+/%&labelG%%_L11G<&F$F3/F*#F+\"\"$/F(,$*&F.F>-F86$,&F>F+* &FEF>FF>/F@%%_L13GF>F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "sols := map(allvalues, [%]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% %solsG7(<&/%\"yG\"\"!/%\"zGF)/%'lambdaG!\"\"/%\"xG\"\"#<&F'F*F,/F0!\"# <&/F0F)F*/F-#F.F1/F(*$F1#\"\"\"F1<&F6F*F7/F(,$F:F.<&F'F6/F-#F.\"\"$/F+ ,$*(F1F \+ " 0 "" {MPLTEXT 1 0 34 "seq(eval(x^2+y^2+z^2, s), s=sols);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(\"\"%F#\"\"#F$#F#\"\"$F%" }}}{PARA 0 "" 0 " " {TEXT -1 117 "So, the smallest sphere containing the ellipsoid has r adius 4 and the largest sphere inside the ellipsoid has radius " } {XPPEDIT 18 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 "." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 264 2 "6." }{TEXT -1 147 " Show that the graph known as the te trahedron is 4-colorable, but not 3-colorable. Show that the dodecahed ron is 3-colorable, but not 2-colorable.\n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "restart:\nwith(Groebner):\nwith(networks):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G := tetrahedron():" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "draw(G);" }}{PARA 13 "" 1 "" {GLPLOT2D 378 378 378 {PLOTDATA 2 "61-%'CURVESG6$7$7$$!+3Q.^?!#>$\"\" \"\"\"!7$$\"+A95`hF*$!\"\"F--%'COLOURG6&%$RGBGF-$\"#5F2F--F$6$7$7$F1$! +:w1-TF*F.F3-F$6$7$7$F+$F-F-F'F3-F$6$7$FBF.F3-F$6$7$F'F " 0 "" {MPLTEXT 1 0 16 "chrompoly(G, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* *%\"tG\"\"\",&F$F%F%!\"\"F%,&F$F%\"\"#F'F%,&F$F%\"\"$F'F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Easy to conclude from this chromatic pol ynomial that the the graph is 4-colorable, but not 3-colorable." }} {PARA 0 "" 0 "" {TEXT -1 65 "Below we apply the Groebner basis method \+ to find the same result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "n := nops(vertices(G));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\" %" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "vars := \{x||(1..n)\}; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%varsG<&%#x1G%#x2G%#x3G%#x4G" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "polys := \{seq(v^3-1, v=va rs)\} union \n \{seq(cat(x,e[1])^2+cat(x,e[1])*cat(x,e[2])\n +cat(x, e[2])^2, e=ends(G))\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&polysG<,, (*$)%#x2G\"\"#\"\"\"F+*&F)F+%#x4GF+F+*$)F-F*F+F+,(*$)%#x1GF*F+F+*&F3F+ %#x3GF+F+*$)F5F*F+F+,(F1F+*&F3F+F)F+F+F'F+,(F'F+*&F)F+F5F+F+F6F+,(F1F+ *&F3F+F-F+F+F.F+,(F6F+*&F5F+F-F+F+F.F+,&*$)F3\"\"$F+F+F+!\"\",&*$)F)FC F+F+F+FD,&*$)F5FCF+F+F+FD,&*$)F-FCF+F+F+FD" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "is_solvable(polys, vars);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "polys := \{seq(v^4-1, v=vars)\} union \n \{seq(cat(x,e[1])^3+cat(x,e [1])^2*cat(x,e[2])\n +cat(x,e[1])*cat(x,e[2])^2+cat(x,e[2])^3, e=ends (G))\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&polysG<,,**$)%#x1G\"\"$ \"\"\"F+*&)F)\"\"#F+%#x2GF+F+*&F)F+)F/F.F+F+*$)F/F*F+F+,*F'F+*&F-F+%#x 4GF+F+*&)F6F.F+F)F+F+*$)F6F*F+F+,*F2F+*&F1F+%#x3GF+F+*&F/F+)F=F.F+F+*$ )F=F*F+F+,*F@F+*&F?F+F6F+F+*&F=F+F8F+F+F9F+,*F'F+*&F-F+F=F+F+*&F)F+F?F +F+F@F+,*F2F+*&F1F+F6F+F+*&F/F+F8F+F+F9F+,&*$)F)\"\"%F+F+F+!\"\",&*$)F /FNF+F+F+FO,&*$)F=FNF+F+F+FO,&*$)F6FNF+F+F+FO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "is_solvable(polys, vars);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G := dodecahedron():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "draw( G);" }}{PARA 13 "" 1 "" {GLPLOT2D 260 260 260 {PLOTDATA 2 "6co-%'CURVE SG6$7$7$$!\"\"\"\"!$!+:w1-T!#>7$$\"+A95`hF-F(-%'COLOURG6&%$RGBGF*$\"#5 F)F*-F$6$7$7$$\"\"\"F*$F*F*7$$!+3Q.^?F-F;F1-F$6$7$F>F'F1-%'POINTSG6#F. -FE6#F'-FE6#F:-FE6#F>-%%TEXTG6$F:Q\"06\"-FE6#7$$\"+j^c5&*!#5$\"+W*p,4$ FW-FN6$FTQ\"1FQ-FE6#7$$\"+V*p,4)FW$\"+CD&y(eFW-FN6$FinQ\"2FQ-FE6#7$$\" +AD&y(eFW$\"+W*p,4)FW-FN6$FcoQ\"3FQ-FE6#7$$\"+Q*p,4$FW$\"+l^c5&*FW-FN6 $F]pQ\"4FQ-FN6$F>Q\"5FQ-FE6#7$$!+U*p,4$FWFU-FN6$FjpQ\"6FQ-FE6#7$$!+FD& y(eFW$\"+T*p,4)FW-FN6$FbqQ\"7FQ-FE6#7$$!+X*p,4)FWFdo-FN6$F\\rQ\"8FQ-FE 6#7$$!+l^c5&*FW$\"+O*p,4$FW-FN6$FdrQ\"9FQ-FN6$F'Q#10FQ-FE6#7$$!+j^c5&* FW$!+W*p,4$FW-FN6$FasQ#11FQ-FE6#7$$!+S*p,4)FW$!+HD&y(eFW-FN6$F[tQ#12FQ -FE6#7$$!+?D&y(eFW$!+Y*p,4)FW-FN6$FetQ#13FQ-FE6#7$$!+M*p,4$FW$!+m^c5&* FW-FN6$F_uQ#14FQ-FN6$F.Q#15FQ-FE6#7$$\"+Y*p,4$FW$!+i^c5&*FW-FN6$F\\vQ# 16FQ-FE6#7$$\"+ID&y(eFW$!+Q*p,4)FW-FN6$FfvQ#17FQ-FE6#7$$\"+Z*p,4)FW$!+ >D&y(eFW-FN6$F`wQ#18FQ-FE6#7$$\"+n^c5&*FW$!+L*p,4$FW-FN6$FjwQ#19FQ-F$6 $7$FTFjpF1-F$6$7$F:FTF1-F$6$7$F:F]pF1-F$6$7$FTFinF1-F$6$7$FinFcoF1-F$6 $7$FasF\\vF1-F$6$7$FbqFetF1-F$6$7$FbqF[tF1-F$6$7$F\\rF_uF1-F$6$7$F\\rF etF1-F$6$7$FdrF_uF1-F$6$7$F.F\\vF1-F$6$7$FjpF[tF1-F$6$7$FetF`wF1-F$6$7 $F_uFjwF1-F$6$7$FinFbqF1-F$6$7$FcoF\\rF1-F$6$7$F]pFdrF1-F$6$7$FfvF`wF1 -F$6$7$F[tFfvF1-F$6$7$F`wFjwF1-F$6$7$F.FjwF1-F$6$7$F\\vFfvF1-F$6$7$Fco F]pF1-F$6$7$F>FasF1-F$6$7$FjpFasF1-F$6$7$FdrF'F1-%*AXESSTYLEG6#%%NONEG " 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "C urve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve \+ 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Cur ve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 3 4" "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" " Curve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curv e 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53 " "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "C urve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "settime := time():" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "chrompoly(G, t);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#**%\"tG\"\"\",&F$F%F%!\"\"F%,&F$F%\"\"#F'F%,F*$)F$\"#7F%)F$\"\"'F%F'*&\")&z3[\"F%)F$\"\"&F%F%*&\")\"Q8Z\"F%)F$\"\"%F%F '*&\")-Oh6F%)F$\"\"$F%F%*&\"(%3#*oF%)F$F)F%F'*&\"(/;v#F%F$F%F%\"'%)fbF 'F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "cputime = (time()-se ttime)*seconds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%(cputimeG,$*&$\"( ()yD\"!\"$\"\"\"%(secondsGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "eval(%%, t=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%+s" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "eval(%%%, t=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 138 "It foll ows that the graph is 3-colorable and not 2-colorable. Using the Groeb ner basis method would result in tremendous computing times. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 " " {TEXT 265 2 "7." }{TEXT -1 45 " The tangent surface of the twisted c ubic in " }{XPPEDIT 18 0 "C^3;" "6#*$)%\"CG\"\"$\"\"\"" }{TEXT -1 34 " is given by the parameterization " }{XPPEDIT 18 0 "[s+t, s*(s+2*t), s ^2*(s+3*t)];" "6#7%,&%\"sG\"\"\"%\"tGF&*&F%F&,&F%F&*&\"\"#F&F'F&F&F&*& )F%F+F&,&F%F&*&\"\"$F&F'F&F&F&" }{TEXT -1 46 ". Prove that it is also \+ given by the equation " }{XPPEDIT 18 0 "-4*x^3*z+3*x^2*y^2-4*y^3+6*x*y *z-z^2 = 0;" "6#/,,*(\"\"%\"\"\"*$)%\"xG\"\"$F'F'%\"zGF'!\"\"*(F+F'*$) F*\"\"#F'F')%\"yGF1F'F'*&F&F'*$)F3F+F'F'F-**\"\"'F'F*F'F3F'F,F'F'*$)F, F1F'F-\"\"!" }{TEXT -1 31 ". Does the same result hold in " }{XPPEDIT 18 0 "R^3;" "6#*$)%\"RG\"\"$\"\"\"" }{TEXT -1 3 "? \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart:\nwith(Groebner):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "polys := [x-(s+t), y-s*(s+2*t), z-s ^2*(s+3*t)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polysG7%,(%\"xG\"\" \"%\"sG!\"\"%\"tGF*,&%\"yGF(*&F)F(,&F)F(*&\"\"#F(F+F(F(F(F*,&%\"zGF(*& )F)F1F(,&F)F(*&\"\"$F(F+F(F(F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "GB := gbasis(polys, plex(s,t,x,y,z)):" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The implicit equation is:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "-GB[1]=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*$)% \"zG\"\"#\"\"\"!\"\"**\"\"'F)%\"xGF)F'F)%\"yGF)F)*(\"\"%F)F'F))F-\"\"$ F)F**(F2F))F.F(F))F-F(F)F)*&F0F))F.F2F)F*\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The result also holds for the real 3-space. If we start with a real point " }{XPPEDIT 18 0 "[x, y, z];" "6#7%%\"xG%\"yG%\"zG" }{TEXT -1 64 " that is a zero of the fi rst polynomial, then there are complex " }{TEXT 288 1 "s" }{TEXT -1 5 " and " }{TEXT 289 1 "t" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "[x , y, z] = [s+t, s*(s+2*t), s^2*(s+3*t)];" "6#/7%%\"xG%\"yG%\"zG7%,&%\" sG\"\"\"%\"tGF+*&F*F+,&F*F+*&\"\"#F+F,F+F+F+*&F*F0,&F*F+*&\"\"$F+F,F+F +F+" }{TEXT -1 21 ". We must prove that " }{TEXT 290 1 "s" }{TEXT -1 5 " and " }{TEXT 291 1 "t" }{TEXT -1 56 " are in fact real. For this, \+ look at the Groebner basis:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "map(collect, GB, [s,t], distributed);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7),,*(\"\"$\"\"\")%\"yG\"\"#F')%\"xGF*F'!\"\"*&\"\"%F') F)F&F'F'**\"\"'F'F,F'%\"zGF'F)F'F-*(F/F'F3F')F,F&F'F'*$)F3F*F'F',,**F/ F'F3F'F+F'F)F'F-*(\"\"&F'F3F'F(F'F'*(F*F'F,F'F7F'F-*&F0F'F,F'F'*&,&*&F *F'F7F'F-*&F*F'F0F'F'F'%\"tGF'F',**&F)F'F3F'F-*&F(F'F,F'F-*(F*F'F+F'F3 F'F'*&,&*&F*F'F(F'F-*(F*F'F,F'F3F'F'F'FBF'F',**&F+F'F)F'F-*&F*F'F(F'F' *&F,F'F3F'F-*&,&F3F-*&F)F'F,F'F'F'FBF'F',*F3F-*(F&F'F)F'F,F'F'*&F*F'F5 F'F-*&,&*&F*F'F+F'F'*&F*F'F)F'F-F'FBF'F',(F)F'*$)FBF*F'F'*$F+F'F-,(F,F -%\"sGF'FBF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "From the second p olynomial follows immediately that " }{TEXT 292 1 "t" }{TEXT -1 53 " i s real. But then the least polynomial implies that " }{TEXT 293 1 "s" }{TEXT -1 14 " is also real." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }