Let $p_1,...,p_n$ be polynomials of degree $t$ in $d$ variables. In the case where $d=n$, the number of zeros of the polynomial is at most $t^n$ by Bezout's Theorem up to an arbitrary perturbation of the polynomials in the supremum norm. In particular, the bound on the number is independent of the coefficients of the polynomial.
In the case where $d>n$, I was able to show $$S=\{x\in[0,1]^d:p_i(x)=0\text{ for }i=1,...,n\}$$ is a manifold of Hausdorff dimension at most $d-n$ under an arbitrarily small perturbation of the polynomials. I am wondering if it is possible to bound the measure of this set independently of the coefficients of the polynomials (just using the degrees of the polynomials, $n$, and $d$).