The polynomial $x^2+y^2$ has an isolated zero at the origin. And so do powers $(x^2+y^2)^n$ of this polynomial. I'm wondering if this is a special property of these real polynomials.

Here's the precise question. Suppose a real polynomial $f(x,y)$ is not divisible by $x^2+y^2$. Is it possible for $f$ to have an isolated zero at the origin?

It seems to me that such a real polynomial would necessarily be either nonnegative or nonpositive in a small neighbourhood of the origin.

More generally, I'm also interested in the case where $f(x,y)$ is real analytic.

EDIT: I'm really delighted by the really comprehensive and diverse reponses I am getting from everybody. Feels like we are in a coffee shop dicsussing mathematics!

Here is what we have gathered. Our basic example of polynomials with isolated zero at the origin is of the form $x^{2n}+y^{2m}$ for positive integers $n$ and $m$. By a linear change of coordinates, this essentially includes examples like $ax^{2n}+by^{2m}$ for positive real coefficients $a$ and $b$.

From this basic example, we can apply perturbion (thanks to Bruno!). Suppose $n\le m$, then we can perturb our basic example add any real analytic function $g(x,y)$ whose lowest degree terms are of degree strictly larger than $m$. The resulting $f+g$ also has an isolated zero.

Does our discussion exhaust the possibilities? Can we say that an polynomial that has an isolated zero is a perturbation of $x^{2n}+y^{2m}$, up to a change of coordinates? Thank you everyone for the lively discussion. Let's charge on!