I have an exam in a few days which includes computing Groebner Basis.
Whilst I have no trouble using Buchberger's algorithm, I am getting confused at smaller 5 mark questions which seemingly skip the algorithm and find the answer quickly.
Here are some examples:
1) Find generating set I ∩ J where $$I = < x^2y + xyz + 1,xy^2 >$$ and $$ J= < x^2 + y^2 +1 > $$
Introducing t by finding $$ N =tI + (1-t)J $$ I compute I ∩ J to be: $$ < tx^2y+txyz+t, txy^2, -tx^2-ty^2+x^2+y^2+t-1 > $$
which is just horrible!
I know from Maple I should be able to reduce this too $$< x^2+y^2-1, t >$$ but I am unsure how. I know how to continue the question from this point.
2) Consider the graph K with vertices VK := {v1, v2, v3} and edges EK := {e12, e13, e23}. Compute a Gr¨obner basis for IK using the lexicographic order with x1 > x2 > x3. Are the vertices of K three-colourable?
Again, I know how and can show that they will be three color-able if the reduced gb is not [1]. I use the verticies and edges to find:
$$< x1^2+x1x2+x2^2, x1^2+x1x3+x3^2, x2^2+x2x3+x3^2, x1^3-1, x2^3-1, x3^3-1 >$$
From hours of staring at this equation, I can reduce it to:
$$< x1^2+x1x2+x2^2, x1^2+x1x3+x3^2, x2^2+x2x3+x3^2, x3^3-1 >$$
I am now stuck how to go further.
Any help would be fantastic!
Thanks.
One can show that $I\cap J = N\cap K[x,y]$ is the first elimination ideal of $N$ in $K[t,x,y]$. By the elimination theorem, using an elimination ordering for $t$ such as the lexicographic ordering with $t>x>y$, compute a Gröbner basis $G$ of $N$. Then $G' = G\cap K[x,y]$ is Gröbner basis of the first elimination ideal $I\cap J$.