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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$. ? Thank you everyone for the lively discussion. Let's charge on! 3 added 864 characters in body The polynomial$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)$with an isolated zero at the origin. Then the lowest degree terms must be of degree$d$at least 1. Collect the lowest degree terms and call the homogeneous polynomial$f_d(x,y)$. I want to claim that the zero set of$f$conincides locally with the zero set of$f_d$, but$x^2+y^4$is a counter example. This perturbation seems needs to be made more precise. I still wish to say that an polynomial that has an isolated zero is essentially of the form$x^{2n}+y^{2m}$. Eg, a change of coordinates will transform$x^2+2y^2$into$x^2+y^2\$. Thank you everyone for the lively discussion. Let's charge on!

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