fourier analysis - level sets of multivariate polynomials

Let $p:mathbb R^n rightarrow mathbb R$ be a polynomial of degree at most $d$. Restrict $p$ to the unit cube $Q=[0,1]^nsubsetmathbb R^n$. We assume that $p$ has mean value zero on the unit cube $Q$:

$$int_Q p(x) dx = 0.$$

For $alpha>0$ consider the sublevel sets of $P$,

$$E_alpha= {xin Q: |p(x)|leq alpha}$$

There are several known estimates for the Lebesgue measure of this set which in some sense or another are uniform over some classes of polynomials. For example, we have that

$$|E_alpha| lesssim min(pd,n) frac{ alpha^{1/d} }{ |p|_{ L^p(Q) }^{1/d} }$$

This particular estimate is due to Carbery and Wright and can be found here.

I'm interested in studying the (induced Lebesgue) measure of the boundary of this set

$$|partial E_alpha|=|{xin Q: |P(x)|=alpha}|$$

Consider first the easy case of dimension $n=1$. Then the set $E_alpha$ is a finite union of closed intervals and the question is trivial. It is obvious that in this case there are at most $O(d)$ intervals so the $0$-dimensional measure of the boundary is $O(d)$.

Now in many variables things will be much more complicated. For example can we say that the set $E_alpha$ has $O(d)$ connected components? Is there an estimate for the measure of the boundary $partial E_alpha$ in terms of $alpha$, $d$ and $n$, assuming (say) that $|p| _ {L^1(Q)}=1$ ?

This question comes up naturally if one tries to study an oscillatory integral with phase $p$ and apply integration by parts (i.e Gauss theorem) imitating the one dimensional method of proving the van der Corput lemma (for example).

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