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. Suppose we have a real analytic function $f(x,y)$ Our basic example of polynomials with an isolated zero at the origin . Then the lowest degree terms must be is of degree $d$ at least 1. Collect the lowest degree terms form $x^{2n}+y^{2m}$ for positive integers $n$ and call the homogeneous polynomial $f_d(x,y)$. I want to claim that the zero set m$. By a linear change of coordinates, this essentially includes examples like $f$ conincides locally with the zero set of ax^{2n}+by^{2m}$ for positive real coefficients $f_d$, but a$ and $x^2+y^4$ is a counter b$.
From this basic example. This perturbation seems needs , we can apply perturbion (thanks to be made more preciseBruno!).
I still wish to 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 essentially a perturbation of the form $x^{2n}+y^{2m}$. Eg, x^{2n}+y^{2m}$, up to a change of coordinateswill transform $x^2+2y^2$ into $x^2+y^2$. ? Thank you everyone for the lively discussion. Let's charge on!
