Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,....,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. My question is, is it plausible to approximate $Z_P$ within $K$ (in the Hausdorff sense, for example) with the zero set of another polynomial $Q(x_1,...., x_n)$ which satisfies the following properties:
$Q$ is "simpler" than $P$ in the sense that a lot more of the coefficients of $Q$ are zero? For example, $x^4 - 1$ is simpler than $x^4 - x^3 + 1$.
The degree of $Q$ is preferably $\leq d$.
If yes, is there is a constructive method for finding $Q$? Any pointer/reference would be highly appreciated!