In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem.

For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\mathbb R^n\to \mathbb R^2$ $$C_a(x_1,\dots, x_n) = (x_1,\ x_1 x_2 + \sum_{i,j\geq 3} a_{i,j} x_i x_j)$$ $C$ stands for "cusp".

Let $\mathcal P$ be the space of polynomial maps $\mathbb R^n\to \mathbb R^2$ of degree at most $2$. This is a real vector space of dimension $N = 1+n + \frac{n(n-1)}{2}$.

Define the subset $\Sigma \subset \mathcal P$ as follows: $p\in \Sigma$ if $\exists A \in GL(n), B \in GL(2), a \in GL(n-2)$ such that $$B \circ p \circ (A x) = C_a(x). $$

In other words $\Sigma$ is the union of the orbits of the maps $C_a$ under the action of $GL(n)\times GL(2)$.

What is the codimension of $\Sigma$ in $\mathcal P$ ? Is there a neat way to see it?

The only thing I can think of is to understand what is trying to compute the dimension of the stabilizer. This should, I believe, be known as cusp singularities have been extensively studied.

More about the motivation. $C_a$ is the local model of cusp singularity. In the space of 2-jets $J^2(M,\mathbb R^2)$ we can define a subset $\mathcal C$ consisting of those 2-jets that are conjugated to a cusp singularity. I am trying to understand what is the codimension of this object.

In trying to prove Cerf's theorem about homotopies between Morse-functions I ended up thinking about the following problem.

For $n>2$, $a= (a_{i,j})\in GL(n-2)$, we define the polynomial map $C_a:\mathbb R^n\to \mathbb R^2$ $$C_a(x_1,\dots, x_n) = (x_1,\ x_1 x_2 + \sum_{i,j\geq 3} a_{i,j} x_i x_j)$$ $C$ stands for "cusp".

Let $\mathcal P$ be the space of polynomial maps $\mathbb R^n\to \mathbb R^2$ of degree at most $2$. This is a real vector space of dimension $N = 1+n + \frac{n(n-1)}{2}$.

Define the subset $\Sigma \subset \mathcal P$ as follows: $p\in \Sigma$ if $\exists A \in GL(n), B \in GL(2), a \in GL(n-2)$ such that $$B \circ p \circ (A x) = C_a(x). $$

In other words $\Sigma$ is the union of the orbits of the maps $C_a$ under the action of $GL(n)\times GL(2)$.

What is the codimension of $\Sigma$ in $\mathcal P$ ? Is there a neat way to see it?

The only thing I can think of is to understand what is trying to compute the dimension of the stabilizer. This should, I believe, be known as cusp singularities have been extensively studied.

More about the motivation. $C_a$ is related to cusp singularities as follows. In the space of 2-jets $J^2(M,\mathbb R^2)$ we can define a subset $\mathcal C$ consisting of those 2-jets that are conjugated to $C_a$. If a map $f:M\to \mathbb R^2$ has 2-jet $j^2f$ transverse to $\mathcal C$ then $f$ is locally (to a point intersecting $\mathcal C$) modelled like a cusp singularity. I am trying to understand what is the codimension of $\mathcal C$.