I am investigating the singularities of a manifold defined as follows, $$ M =\{x \in \mathbb{R}^n \, |\, g_1(x)=u_1, \cdots, g_p(x)=u_p\} $$ where $g_i:\mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\cdots,p$ are homogeneous polynomials and $u_i$ are constants and $n > p$. The partial derivatives $\frac{\partial g_i}{\partial x_j}$ are linear independent and homogeneous polynomials for all $1\leq i\leq p$ and $1\leq j \leq n$.
Let $Sing^{(n)}(g_1, \cdots, g_p)$ be the locus on $\mathbb{R}^n$ where the gradients $\nabla g_i$ are linearly dependent, more precisely $$ Sing^{(n)}(g_1, \cdots, g_p) =\{x \in \mathbb{R}^n| \mbox{It exist } \{\alpha_i\}_{i=1}^p \subset \mathbb{R} \mbox{ such that } \sum_i \alpha_i \nabla g_i(x)=0\} $$ I am interested to bound the dimension of $Sing^{(n)}(g_1, \cdots, g_p)$. One trivial bound for $dim(Sing^{(n)}(g_1, \cdots, g_p))$ is $n-1$, but I think that it is possible to find better upper bounds due to the linear independence of $\frac{\partial g_i}{\partial x_j}$
Some of you have an idea how to tackle this problem?
Thanks in advance.