Construction of universal Thom-Whitney-a stratifications, their functoriality and Sard-type Theorem for singular varieties

{\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {\it bundle} $G_F \subset T^{*}K^n$. We restrict $G_F$ over the critical points $Sing(F)$ of $F$ in $F^{-1}(0)$ and partition $Sing(F)$ into {\it 'quasistrata'} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications 'near' $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the quasistrata (e. g. when $l=1$). Also, the latter are the orthogonal complements over an irreducible component $S$ of a quasistratum only if $S$ is {\bf universal} for the class of {T-W-a} stratifications, meaning that for any $\{S_j'\}_j$ in the class, $\Sing (F) = \cup_j S'_j$, there is a component $S'$ of an $S_j'$ with $S\cap S'$ being open and dense in both $S$ and $S'$. {\bf Results.} We prove that T-W-a stratifications with only universal strata exist iff all fibers of $G_F$ are the orthogonal complements to the respective tangent spaces to the quasistrata, and then the partition of $\Sing(F)$ by the latter yields the coarsest {\it universal T-W-a stratification}. The key ingredient is our version of {\bf Sard-type Theorem for singular spaces} in which a singular point is considered to be noncritical iff nonsingular points nearby are 'uniformly noncritical' (e. g. for a dominating map $F: X \to Z$ meaning that the sum of the absolute values of the $l\times l$ minors of the Jacobian matrix of $F$, where $l = \dim (Z)$, not only does not vanish but, moreover, is separated from zero by a positive constant).