MCS 563 Friday 27 August 2004 Illustration of Groebner basis computation in Maple 9

restart;

We use the example on page 26 of the textbook.

p1 := x^2 + 4*x*y + 4*y^2 - 4;

NiM+SSNwMUc2IiwqKiRJInhHRiUiIiMiIiIqJkYoRipJInlHRiVGKiIiJSokRixGKUYtISIlRio=

p2 := 4*x^2 - 4*x*y + y^2 - 4;

NiM+SSNwMkc2IiwqKiRJInhHRiUiIiMiIiUqJkYoIiIiSSJ5R0YlRiwhIiUqJEYtRilGLEYuRiw=

The polynomials in the system are products of linear polynomials:

factor(p1);

NiMqJiwoSSJ4RzYiIiIiSSJ5R0YmIiIjRilGJ0YnLChGJUYnRihGKSEiI0YnRic=

factor(p2);

NiMqJiwoSSJ4RzYiIiIjSSJ5R0YmISIiRiciIiJGKiwoRiVGJ0YoRikhIiNGKkYq

with(Groebner);

NiM3OEktR0JfSW50ZXJuYWxzRzYiSSpNdWxNYXRyaXhHRiVJKVNldEJhc2lzR0YlSSpmZ2xtX2FsZ29HRiVJJ2diYXNpc0dGJUknZ3NvbHZlR0YlSStoaWxiZXJ0ZGltR0YlSSxoaWxiZXJ0cG9seUdGJUkuaGlsYmVydHNlcmllc0dGJUktaW50ZXJfcmVkdWNlR0YlSSppc19maW5pdGVHRiVJLGlzX3NvbHZhYmxlR0YlSSpsZWFkY29lZmZHRiVJKGxlYWRtb25HRiVJKWxlYWR0ZXJtR0kqcHJvdGVjdGVkR0Y0SShub3JtYWxmR0YlSS9wcmV0ZW5kX2diYXNpc0dGJUkncmVkdWNlR0YlSSZzcG9seUdGJUkqdGVybW9yZGVyR0YlSSp0ZXN0b3JkZXJHRiVJKXVuaXZwb2x5R0Yl

?Groebner;

triangular_basis := gbasis([p1,p2],plex(x,y));

NiM+STF0cmlhbmd1bGFyX2Jhc2lzRzYiNyQsKCokSSJ5R0YlIiIjISUrNSIkVyIiIiIqJEYpIiIlIiREJywoKiRGKSIiJCIkRCJGKSEkayJJInhHRiUiI1s=

total_degree_basis := gbasis([p1,p2],tdeg(x,y));

NiM+STN0b3RhbF9kZWdyZWVfYmFzaXNHNiI3JSwoKiZJInhHRiUiIiJJInlHRiVGKiIjPyokRisiIiMiIzohIzdGKiwoKiRGKUYuIiImRi1GMyEiKUYqLCgqJEYrIiIkIiREIkYrISRrIkYpIiNb

?SetBasis;

ns,rv := SetBasis(total_degree_basis,tdeg(x,y));

NiM+NiRJI25zRzYiSSNydkdGJjYkNyYiIiJJInlHRiZJInhHRiYqJEYrIiIjLUkmVEFCTEVHSSpwcm90ZWN0ZWRHRjE2IzcmL0YqRiovRi0iIiUvRiwiIiQvRitGLg==

My := MulMatrix(y,ns,rv,total_degree_basis,tdeg(x,y));

NiM+SSNNeUc2Ii1JJ1JUQUJMRUdGJTYlIipLWmFPIi1JJ01BVFJJWEdGJTYjNyY3JiIiISIiIkYvRi83JkYvRi9GL0YwNyYjIiIkIiImRi9GLyMhIiQiIiU3JkYvIyIkayIiJEQiIyEjW0Y8Ri9JJ01hdHJpeEc2JEkqcHJvdGVjdGVkR0ZBSShfc3lzbGliR0Yl

y_values := LinearAlgebra[Eigenvalues](My);

NiM+SSl5X3ZhbHVlc0c2Ii1JJ1JUQUJMRUdGJTYlIionKmYwTyItSSdNQVRSSVhHRiU2IzcmNyMjIiIjIiImNyMjISIjRjE3IyMiIidGMTcjIyEiJ0YxJkknVmVjdG9yRzYkSSpwcm90ZWN0ZWRHRj5JKF9zeXNsaWJHRiU2I0knY29sdW1uR0Yl

seq(subs(y=y_values[k],triangular_basis),k=1..LinearAlgebra[Dimension](y_values));

NiY3JCIiISwmIyEkKUciIiYiIiJJInhHNiIiI1s3JEYkLCYjIiQpR0YoRilGKkYsNyRGJCwmIyIjJypGKEYpRipGLDckRiQsJiMhIycqRihGKUYqRiw=

TTdSMApJNlJUQUJMRV9TQVZFLzEzNjU0NDczMlguJSlhbnl0aGluZ0c2IjYiW2dsISIhISEhIyEiJSIlISFFXFtsJzYkIiIlIiIjIyIkayIiJEQiCjYkIiIkRikjISIkRik2JEYvIiIiI0YvIiImNiRGKUYvIyEjW0YtNiRGM0YqRjM2JEYqRilGM0YmCg==TTdSMApJNlJUQUJMRV9TQVZFLzEzNjA1NTk5NlgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiUiJSMiIiMiIiYjISIjRikjIiInRikjISInRilGCiYK