I need to calculate a Grobner basis of $$I=(u-x^2,v-y^2,w-xy)$$ with lexicographic order $x>y>u>v>w$.
I am trying to proceed with Buchberger's algorithm, but i probably didn't quite understand how to apply the multivariate division algorithm.
Let $f_1:=u-x^2$, $f_2:=v-y^2$, $f_3:=w-xy$.
Denote by $g_i$ the leading term of $f_i$ with respect to the given ordering $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; g_1:=x^2$, $g_2:=y^2$, $g_3:=xy$.
Denote by $a_{ij}$ the least common multiple of $g_i$ and $g_j$.$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;a_{1,2}=x^2y^2$, $a_{1,3}=x^2y$, $a_{2,3}=xy^2$
Choose two polynomials in G and let $S_{ij} = (a_{ij} / g_i) f_i − (a_{ij} / g_j) f_j. \;\;\;$I chose $S_{1,2}=y^2(u-x^2)-x^2(v-y^2)$
But now, from multivariate division algorithm of $S_{ij}$ by $f_1,f_2,f_3$ always gives $r=0$. What am I doing wrong?
We calculate $S_{1,3}=\frac{x^2y}{-x^2}f_1-\frac{x^2y}{-xy}f_2=xw-yu$ . Now by the lexicographic order the leading term of $S_{1,3}$ is given by $xw$ but this is not divisible by the leading terms of any of the three given polynomials. Now we initiate the multivariate division algorithm with the following data:
1)$r^{(0)}=0$, $p^{(0)}=S_{1,3}$
Now the leading term of $p$ is given by $xw$ and this is not divisible by any of the leading terms of the $f_i$ so following the algorithm we have
2)$p^{(1)}=p^{(0)}-lt(p^{(0)})=-yu$ and $r^{(1)}=r^{(0)}+lt(p^{(0)})=xw$
Now we notice again that the leading term of $p^{(1)}$ is not divisible by any of the leading terms of the $f_i$ so doing another step in the algorithm we obtain:
3)$p^{(2)}=p^{(1)}-lt(p^{(1)})=-yu+yu=0$ and $r^{(2)}=r^{(1)}+lt(p^{(1)})=xw-yu=S_{1,3}$
Now our $p^{(2)}=0$ hence we stop the algorithm and add the rest $r=S_{1,3}$ to the divisors and start from scratch.
Note that in the multivariate division divisibility only depends on the leading terms. I hope this clarifies some of your problems! The full Gröbner basis is given by $[x^2 - u, xy - w, xv - yw, xw - yu, y^2 - v, uv - w^2]$. Just adding this to check the results of your further calculations :)
Lg Mo