restart; with(Groebner); # SOLUTION OF PROBLEM ONE IN REVIEW SESSION NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl f:=x^4+y^2+z^2-1; NiM+SSJmRzYiLCoqJEkieEdGJSIiJSIiIiokSSJ5R0YlIiIjRioqJEkiekdGJUYtRiohIiJGKg== g:=(x-1)^2+(y-1)^2+(z-1)^2; NiM+SSJnRzYiLCgqJCwmSSJ4R0YlIiIiISIiRioiIiNGKiokLCZJInlHRiVGKkYrRipGLEYqKiQsJkkiekdGJUYqRitGKkYsRio= K:=[g,diff(g-l*f,x),diff(g-l*f,y),diff(g-l*f,z)]; NiM+SSJLRzYiNyYsKCokLCZJInhHRiUiIiIhIiJGKyIiI0YrKiQsJkkieUdGJUYrRixGK0YtRisqJCwmSSJ6R0YlRitGLEYrRi1GKywoRipGLSEiI0YrKiZJImxHRiVGK0YqIiIkISIlLChGMEYtRjVGKyomRjdGK0YwRitGNSwoRjNGLUY1RisqJkY3RitGM0YrRjU= L:=gbasis(K,plex(l,x,y,z)); NiM+SSJMRzYiNyYsNCokSSJ6R0YlIiImISUhMyIqJEYpIiIlIiVoOiokRikiIiQhJXE5KiRGKSIiIyIkKCkpRikhJD8kIiNhIiIiKiRGKSIiKSIjOyokRikiIighJEciKiRGKSIiJyIkIVssJkYpISIiSSJ5R0YlRjcsNEkieEdGJSIkOiJGLyEnZHc3RjIiJncuKkYpISYiSFAiJThyRjdGOyElL0ZGPiImLyQ9RighJiNmZUYsIidPLjYsNEkibEdGJUY2RjshIztGPiIkRyJGKCEkIVtGLCIlITMiRi8hJWg6RjIiJXE5RikhJCgpKSIkbSNGNw== factor(L); NiM3JiomLDAqJEkiekc2IiIiJyIjOyokRiciIiYhIycqKiRGJyIiJSIkcyMqJEYnIiIkISRTJSokRiciIiMiJDQlRichJDcjIiNhIiIiRjksJkYnRjkhIiJGOUY1LCZGJ0Y7SSJ5R0YoRjksNEkieEdGKCIkOiJGMSEnZHc3RjQiJncuKkYnISYiSFAiJThyRjkqJEYnIiIoISUvRkYmIiYvJD1GKyEmI2ZlRi4iJ08uNiw0SSJsR0YoRjhGRSEjO0YmIiRHIkYrISQhW0YuIiUhMyJGMSElaDpGNCIlcTlGJyEkKCkpIiRtI0Y5 assign({solve(L[2],{y}), solve(L[3],{x})}); rootlist:=[evalf(solve(L[1]))]; NiM+SSlyb290bGlzdEc2IjcqJCIiIiIiIUYnXiQkIitGMTRtOCEiKiQiK3cuYDI2ISM1XiQkIismM2VUZyVGMCQiK3AsT2dhRjBeJCQiK2xOXHQ2Ri0kIisqR0pYWSJGLV4kRjckISsqR0pYWSJGLV4kRjIkIStwLE9nYUYwXiRGKyQhK3cuYDI2RjA= for z in rootlist do print(['x'=x,'y'=y,'z'=z]); od; NiM3JS9JInhHNiIkIikvKys1ISIoL0kieUdGJiQiIiIiIiEvSSJ6R0YmRiw= NiM3JS9JInhHNiIkIikvKys1ISIoL0kieUdGJiQiIiIiIiEvSSJ6R0YmRiw= NiM3JS9JInhHNiJeJCQiJ3NMJSkhIickIidKeF5GKi9JInlHRiZeJCQiK0YxNG04ISIqJCIrdy5gMjYhIzUvSSJ6R0YmRi8= NiM3JS9JInhHNiJeJCQiKCMpeUYjISIoJCEpNigzaighIikvSSJ5R0YmXiQkIismM2VUZyUhIzUkIitwLE9nYUYzL0kiekdGJkYw NiM3JS9JInhHNiJeJCQhJztyNSEiJiQiJ2ZgQyEiJy9JInlHRiZeJCQiK2xOXHQ2ISIqJCIrKkdKWFkiRjMvSSJ6R0YmRjA= NiM3JS9JInhHNiJeJCQhJztyNSEiJiQhJ2ZgQyEiJy9JInlHRiZeJCQiK2xOXHQ2ISIqJCErKkdKWFkiRjMvSSJ6R0YmRjA= NiM3JS9JInhHNiJeJCQiKCMpeUYjISIoJCIpNigzaighIikvSSJ5R0YmXiQkIismM2VUZyUhIzUkIStwLE9nYUYzL0kiekdGJkYw NiM3JS9JInhHNiJeJCQiJ3NMJSkhIickISdKeF5GKi9JInlHRiZeJCQiK0YxNG04ISIqJCErdy5gMjYhIzUvSSJ6R0YmRi8= restart; F:=(x^2+y^2-4)*(x^2+y^2-1)+(x-3/2)^2+(y-3/2)^2; # SOLUTION OF PROBLEM TWO NiM+SSJGRzYiLCgqJiwoKiRJInhHRiUiIiMiIiIqJEkieUdGJUYrRiwhIiVGLEYsLChGKUYsRi1GLCEiIkYsRixGLCokLCZGKkYsIyEiJEYrRixGK0YsKiQsJkYuRixGNEYsRitGLA== G:=[diff(F,x),diff(F,y)]; NiM+SSJHRzYiNyQsKiomSSJ4R0YlIiIiLCgqJEYpIiIjRioqJEkieUdGJUYtRiohIiJGKkYqRi0qJiwoRixGKkYuRiohIiVGKkYqRilGKkYtRilGLSEiJEYqLCoqJkYvRipGK0YqRi0qJkYyRipGL0YqRi1GL0YtRjRGKg== with(Groebner); gbasis(G,plex(x,y)); NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl NiM3JCwoKiRJInlHNiIiIiQiIilGJiEiKSEiJCIiIiwmRiYhIiJJInhHRidGLA== # SOLUTION OF PROBLEM FOUR restart; with(Groebner); NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl F:=[x^2+y^2+z^2-1,x^2+y^2+z^2-2*x,2*x-3*y-z]; NiM+SSJGRzYiNyUsKiokSSJ4R0YlIiIjIiIiKiRJInlHRiVGKkYrKiRJInpHRiVGKkYrISIiRissKkYoRitGLEYrRi5GK0YpISIjLChGKUYqRi0hIiRGL0Yw with(Groebner); NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl L:=gbasis(F,plex(x,y,z)); NiM+SSJMRzYiNyUsKCokSSJ6R0YlIiIjIiNTRikhIikhI0IiIiIsKCEiIkYuSSJ5R0YlIiIkRilGLiwmRjBGLkkieEdGJUYq assign({solve(L[2],{y}), solve(L[3],{x})}); rootlist:=[solve(L[1])]; NiM+SSlyb290bGlzdEc2IjckLCYjIiIiIiM1RikqJCIjRSNGKSIiIyMiIiQiIz8sJkYoRilGKyMhIiRGMQ== for z in rootlist do print(['x'=x,'y'=y,'z'=z]); od; NiM3JS9JInhHNiIjIiIiIiIjL0kieUdGJiwmIyIiJCIjNUYoKiQiI0VGJyMhIiIiIz8vSSJ6R0YmLCYjRihGL0YoRjAjRi5GNA== NiM3JS9JInhHNiIjIiIiIiIjL0kieUdGJiwmIyIiJCIjNUYoKiQiI0VGJyNGKCIjPy9JInpHRiYsJiNGKEYvRihGMCMhIiRGMw==