Ideal of $\mathbb{C}[x,y]$ not generated by two elements

Question

Consider the ring of polynomials in two variables $\mathbb{C}[x,y]$. Show that the ideal $\langle xy^3, x^2y^2, x^3y\rangle$ cannot be generated by two elements.

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Until now, I assumed by contradiction that $\langle xy^3, x^2y^2, x^3y\rangle=\langle g, h \rangle$ for some $g,h \in \mathbb{C}[x,y]$. Then, we have that:

$g=xy^3a + x^2y^2b + x^3yc$

$h=xy^3d + x^2y^2e + x^3yf$

for $a, b, c, d, e, f$ in the polynomial ring. However, I'm not sure if this is the correct approach as I'm not sure on how to proceed. I was thinking that maybe one can write an element in $\langle xy^3, x^2y^2, x^3y\rangle$ as the square of a sum. What would be the best approach to do this?

Thanks for the help.

Answer 1

Set $I=\langle xy^3, x^2y^2, x^3y\rangle$ and $\mathfrak m=\langle x, y\rangle$.

$\dim I/\mathfrak mI=3$

Just show that the residue classes of the generators of $I$ are linearly independent over $K$, that is, if $axy^3+bx^2y^2+cxy^3\in mI$ with $a,b,c\in K$ then $a=b=c=0$. This is obvious since the generators of $\mathfrak mI$ are homogeneous of degree $5$.

If $\dim I/\mathfrak mI=3$ then the minimal number of generators of $I$ is $3$.

This is very well explained here.