On the codimension of the singular locus

Let $k$ be a field and $V$ an $k$-vector space. For a family $\bar P={ P_i}_{1\leq i\leq c}, $ofpolynomialson$V$,wedenoteby$\mathbb X _{\bar P}\subset V$thesubschemedefinedbytheidealgeneratedby$ \bar P$. We show the existence of $\gamma (c,d)$ such that the varieties ${\mathbb{X}}_{\bar{P}}$ are smooth outside of codimension $m$, if deg$(P_i)\le d$ and rank (strength) $r_{nc}(\bar{P})\ge \gamma (d,c)(1+m{)}^{\gamma (d,c)}$.