Polynomials vanishing on subsets of $\mathbb{R}^2$ Let $\mathcal{S}\subset\mathbb{R}^2$ such that every point in the real plane is at most at distance $1$ from a point in $\mathcal{S}$. Is it true that if $P\in\mathbb{R}[X,Y]$ is a polynomial that vanishes on $\mathcal{S}$, then $P=0$?

Following the suggestions in comments, let's expand $P$ into a sum of homogeneous polynomials: $P=P_0+\dots+P_n$, with each $P_k$ homogeneous of degree $k$, and $P_n$ is not identically zero. Pick a closed ball $B\subset \mathbb R^n$ on which $P_n$ is nonzero (this is possible because the zero set of $P_n$ has empty interior). For sufficiently large $\lambda>0$ we have $$\inf_{\lambda B}|P_n|>\sup_{\lambda B}\sum_{k}|P_k|$$  because  the left hand side  is proportional to $\lambda^n$ while the right hand side is $O(\lambda^{n-1})$. Hence $P\ne 0$ on $\lambda B$, and the radius of this ball can be arbitrarily large. 
