Prove that ideal generated by.... Is a monomial ideal

Question

Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated..

Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by $X^2(1+Y^3), Y^3(1-X^2), X^4$ and $ Y^6$ is a monomial ideal.

@DanieleA yeah I know Grobner Bases but not sure how to use this to prove this? — Mathsstudent147, May 20 '13 at 18:47
@TobiasKildetoft the definition I have is' I is a monomial ideal if it has a basis formed of monomials' — Mathsstudent147, May 20 '13 at 19:01
With Grobner bases, you can use the fact that an ideal is monomial if and only if its reduced Grobner basis (w.r.t. any ordering) is monomial. Maybe in this case you can compute a Grobner basis easily — Daniele A, May 21 '13 at 14:11

score 6 · Answer 1 · answered Jan 05 '14 at 18:50

6

Hint. $(X^2(1+Y^3), Y^3(1-X^2), X^4, Y^6)=(X^2,Y^3)$.

answered Jan 05 '14 at 18:50

user209116 · Answer 2 · 2015-01-20T02:39:10.207

0

Let $$\alpha_1=x^2+x^2y^3, \quad\alpha_2=x^4, \quad \beta_1=y^3-x^2y^3, \quad\beta_2=y^6.$$ Then, $(\alpha_1,\alpha_2,\beta_1,\beta_2)\subset (x^2,y^3)$ is easy. The containment $\supset$ is due to the following expressions: $$x^2=\alpha_1-x^2\beta_1-y^3\alpha_2$$ $$y^3=\beta_1+y^3\alpha_1-x^2\beta_2.$$

edited Jan 20 '15 at 02:39

answered Jan 20 '15 at 02:34

user209116

1

Prove that ideal generated by.... Is a monomial ideal

2 Answers2