Let $f,g \in \mathbf {C}[x\_1, \ldots, x\_n]\subseteq \mathbf {C}[x\_1, \ldots, x\_{n+1}]$ be two polynomials with complex coefficients and suppose that there exists $h\_1 \in \operatorname{GL}\_{n+1}(\mathbf C)$ such that $h\_1 \cdot f=g$. Does there exist $h\_2 \in \operatorname{GL}\_n(\mathbf C)$ such that $h\_2\cdot f=g$?


This is a toy example of a more general question I'm interested in: given a family of representations $\rho\_n: \operatorname{GL}\_n \to \operatorname{End}(V\_n)$ such that $V\_n$ embeds naturally in $V\_{n+1}$ (e.g., $V\_n=\Lambda^r(\mathbf C^n)$), what can be said about the $\operatorname{GL}\_{n+1}$-orbits of elements of $V\_n$?



GL(n+1)- vs. GL(n)-orbits
Yes, this is true. Let me consider the case when there is no linear change of coordinates $(x\_1,...,x\_n)$ such that $g=g(x\_1,...,x\_{n-1})$. In this case it is clear that the vector $v=h\_1(0,...,1)$ does not belong to the plane $(x\_{n+1}=0)\subset \mathbb C^{n+1}$. Hence we can find a linear transformation $h\_3$ of $\mathbb C^{n+1}$ that lives invariant all lines in $\mathbb C^{n+1}$ parallel to $v$ and sends the plane $h\_1((x\_{n+1}=0))$ to $(x\_{n+1}=0)$. This transformation leaves $h\_1(f)$ invariant. Finally chose $h\_2$ to be the restriction of $h\_3\circ h\_1$ to the plane $x\_{n+1}=0$.


The general case is similar. We just need to play with the largest subgroups in the group of translations of $\mathbb C^n$ that leave $f$ and $g$ invariant (respectively) - such groups are unique (and have the same dimension $0 \le dim\le n$). 


