Suppose $f:\mathbb{R}^d\to\mathbb{R}$ is a polynomial of degree $\ell$. Then, the number of connected components of its zero set $\{a\in\mathbb{R}^d : f(a) = 0\}$ is bounded by roughly $\ell^{d}$. I've seen this result attributed to Warren, Milnor-Thom, and stated to be a consequence of Bezout's theorem (a concrete reference would be much appreciated, especially one that derives this using Bezout's theorem).
Onto my main question: suppose $f$ is known to only have $k$ terms. Is there a better bound on the number of connected components of the zero set of $f$ that depends on $k$? I suspect that there might be a reference that answers this, but I don't know much about this area and would appreciate any suggestions on where to look. Thanks in advance!
For $d=1$, you can use Descartes' rule of signs.
The number of positive roots is upper bounded by $k$ (since there are at most $k$ sign changes).
The number of negative roots is the number of positive roots of $f(-x)$ which is at most $k$ for the same reason.
Thus, we get a total bound of $2k+2$ (there are $2k+1$ real roots in total (including possibly $0$)).