# Help with a Groebner basis

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?

• (u-x^2)*y^2+u*(v-y^2)-(w+x*y)*(w-x*y) Commented Dec 4, 2020 at 15:53

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

• I proceeded as suggested: I arrived at $$(u-x^2,v-y^2,w-xy,xw-yu,xv-yw)$$ So, $$g_1=-x^2\;g_2=-y^2\;g_3=-xy\;g_4=xw\;g_5=xv$$ Then I calculate the different $a_{ij}$, in particular $$a_{4,5}=xvw$$. Then $$S_{4,5}=w^2y-uvy$$ and for multivariate division algorithm I should add $w^2y-uvy$ in Groebner basis. But I want only $w^2-uv$.....What is wrong? Commented Dec 4, 2020 at 21:44
• There is nothing wrong with your answer! In general a groebner basis is NOT unique. We only obtain uniqueness is we demand the groebner basis to be reduced(see for example homepage.univie.ac.at/dietrich.burde/papers/… Prop 2.5.8 for a proof of the uniqueness and Def 2.5.7 for the term "reduced"). After looking a bit into the script you will see, that the basis i gave is indeed reduced and yours is not. That does not mean your answer is wrong. The basis you give is still a gröbner basis, just not reduced. Commented Dec 4, 2020 at 22:14
• It's just that most computer algebra programms compute the reduced version and since i calculated mine using sagemath(if you are interested in using computer algebra programms see for example sagemath.org) it looked different from yours. Commented Dec 4, 2020 at 22:14
• I hope that clarifies your concerns! Lg Mo Commented Dec 4, 2020 at 22:15
• Ah! Ok!!!! Thank you! Commented Dec 4, 2020 at 22:39