I am currently revising for my exams and working on questions about monomial ideals and came across this question.
Let I be the ideal of $\mathbb{R}[x,y]$ generated by all polynomials of the form $x^iy^j$ with $i,j \in \mathbb{Z}$, $i>0$ and $j \geq (i − 2)^2$. Find a finite set $G \in \mathbb{R}[x,y]$ such that $I = <G>$.
I'm not too sure how to go about this, I was thinking about manipulating the inequalities and subbing $j$ into the monomial but can't seem to find a finite set.
Any help would be greatly appreciated
You can write down the first several monomials:
$$i=1, x^1y^{1+k}, k\geq 0 \implies xy\\ i=2, x^2y^k, k\geq 0 \implies x^2, x^2y, x^2y^2,...\\ i=3, x^3 y^{1+k}, k\geq 0 \implies x^3y,...\\ i=4, x^4 y^{4+k}, k\geq 0 \implies x^4y^4, ...$$
From this point on, it is very clear that every monomial is divisable by some previous one. See if you can find a finite bases from those.