Sufficient conditions on a semialgebraic set for being a manifold

Question

Let $A = \{x \in \mathbb{R}^n \ | \ \forall i \in \{1,...,m\}, P_i(x) \geq 0\}$ be a semialgebraic subset of $\mathbb{R}^n$ (for some $n,m \in \mathbb{N}^\ast$).

Are there some sufficient conditions on the $(P_i)_{i =1}^m$ for $A$ being a topological submanifold of $\mathbb{R}^n$ ? A differentiable submanifold of $\mathbb{R}^n$ ?