Let I, J ideals. Are they equal?

Question

Let

$$I= \langle 11x^5y+7xy^6+9,8xy^4+6xy+9 \rangle$$

$$J= \langle 7x^5y^2+17x^2y^5+29,13xy^4+62xy^3+19 \rangle$$

ideals. Examine whether those two ideals are equal.

By seeing their 3D plots I think they are not equal, but I dont know how to show it...

Any help appreciated!

Answer 1

You can use Buchberger's algorithm to prove that those ideals are not equal. Pick any monomial order on $k[x,y]$ and construct the reduced Groebner basis for each of those ideals. Since reduced Groebner basis is unique for every ideal fixed, this approach will give a straightforward answer.

For, example, for grlex ($x \succ y$) ordering: $$I = \langle 7y^5+11x^4-8y^3-6, \; 8xy^4+6xy+9, \; 88x^5-64xy^3-42xy^2-48x-63y \rangle,$$ while: $$J = \langle 133x^4+323xy^3-377y^2-1798y, \;13xy^4+62xy^3+19, \;4901y^6+46748y^5+111476y^4+2527x^3+6137y^3 \rangle.$$ So $I \neq J$.

By the way, I think that this question should be tagged as a homework.