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

$\int_Q f(x) dx = 0.$

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

$$E_\alpha= \{x\in Q: |p(x)|\leq \alpha\}$$

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

$$|\partial E_\alpha|=|\{x\in Q: |P(x)|=\alpha\}| $$

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).