I have a family of multi-variate polynomials, with bounded degree and a bound on the norm of the coefficients, say for example, all multi variate polynomials of the form: $\{ a_1 xy + a_2x + a_3xyz: |a_1|,|a_2|,|a_3| \le 1\}$.
Each polynomial defines a surface in Euclidean space by considering its common roots i.e. $A_p = \{(x,y,z): p(x,y,z)=0\}$. We assume that $p=0$ is not in our family, so all surfaces are proper hyper surfaces.
Given that the degree of the polynomials is uniformly bounded and the size of the coefficients is uniformly bounded, can I have a uniform upper bound on the surface area (Haussdorf measure) of the surfaces when intersected with a compact domain? i.e. bound on
$\sup_p \mathcal{H}(A_p\cap K)$
A theoretical bound will suffice.
Here is a general answer: Let $S\subset R^n$ be a $k$-dimensional real-algebraic submanifold of degree $\le d$, meaning that every generic affine subspace of dimension $n-k$ intersects $S$ is at most $d$ points. Then for every round ball $B$ of radius $r$, we have $$ Vol_{k}(S\cap B)\le C(n,k,d)r^n, $$ where $C$ is some universal function independent of $r$ and $S$. This is an application of Crofton's formula, see Theorem 4.3 in
A. Paiva, E. Fernandes, "Gelfand transforms and Crofton formulas", http://www.math.poly.edu/~alvarez/pdfs/crofton.pdf (I am pretty sure one can find it elsewhere too, this is just a free source that I found).
Thus, in your setting, you need only to bound the degree of your surface and not the coefficients.