So Gröbner bases lead to nothing new for the Beal polynomial $x^a + y^b - z^c$?

Question

I'm learning about Gröbner bases. And $f(x,y,z) = x^a + y^b - z^c$, is a single monic polynomial in any monomial ordering, $I = (f)$ has Gröbner basis $\{f\}$. So there's nothing interesting to look at. What am I missing?

Thanks.

Answer 1

Any principal ideal is going to be its own Groebner basis. Let $lt(g)$ be the leading term of a polynomial $g$. The leading term of an element in $(f)$ is going to be $lt(f)r$ for some monomial $r$ since we are looking at polynomials over fields, which are integral domains. All of the interesting things we can do with Groebner bases, like testing if a polynomial is in a certain ideal, are easy when we have a principal ideal. For example, a polynomial is in $(f)$ if the remainder of polynomial division is $0$.