Say I have polynomials $p_1,p_2,\dots,p_m$ in $\mathbb{R}^n$ (ie. over $n$ variables), each of degree $d$. Is there an upper bound on the number of "regions" created by the surfaces $p_i = 0$? Let's say $n$-dimensional regions, though I am ultimately interested in a bound on the number of $k$-dimensional regions for all $k$.

I'm hoping for something polynomial in $m$, and at most exponential in $n$ and $d$. For instance, when $d=1$, this is just an arrangement of hyperplanes, and the number of regions is bounded by $m^n$.

Pointers to literature, proof ideas, etc would all be of help.