Let $R = \mathbb{R}[X,Y]$ and let $\succcurlyeq$ be a monomial ordering on $M(X,Y)$. Let $I$ be the ideal $\left<X^3, X^2Y,XY^2,Y^3\right>$. Why can't $I$ have a Grobner basis consisting of $3$ elements?
I have no idea where to start here. Is it something to do with the Buchberger algorithm or?
Any help would be appreciated!
The ideal $I$ is a monomial ideal this means the generators form a Gröbner basis. No generator is divisible by any of the other generators. Thus $\{X^3, X^2Y, XY^2, Y^3\}$ is a minimal Gröbner basis.
A minimal monomial basis $B = \{b_1, \dots ,b_k\}$ of a monomial ideal $I$ is unique. Can you proof this?
Hint: Assume there exists a second minimal monomial basis $B'$. Use the fact that a monomial $m$ is in $I$ if and only if $m$ is divisible by $b_i$ for some $i \in \{1, \dots, k\}$ to show that $B \subseteq B'$. By interchanging $B$ and $B'$ you then get the equality of the bases.