Which complex polynomials in 3 variables are $GL_3(\Bbb{C})$ invariant?

Question

A polynomial $p(x,y,z) \in \Bbb{C}[x,y,z]$ is $GL_3(\Bbb{C})$-invariant if $$ \forall \sigma \in GL_3(\Bbb{C}): p(\sigma(x,y,z)) = p(x,y,z).$$

How to characterize the set of $GL_3(\Bbb{C})$ invariant polynomials ?

Answer 1

Only the constant polynomials are $GL_3(\Bbb{C})$-invariant. Let us take $p(x,y,z)$ and write it in powers of $x$: $$ p(x,y,z) = \sum_{i=0}^n x^i Q_i(y,z),$$ where $Q_n$ is non zero. Let $\sigma \in GL_3(\Bbb{C})$ be defined by $$ \sigma(x,y,z) := (x+y,y,z).$$ Now $$ p \circ \sigma (x,y,z) = p(x,y,z) + n x^{n-1} y Q_n(y,z) + \mbox{ lower order terms in } x.$$ But since $Q_n$ is not zero, this cannot equal $p$. Thus we cannot have a polynomial which involves $x$, and is invariant even under $\sigma$, let alone under all of $GL_3(\Bbb{C})$. The same holds, of course, for all other variables, and so the only invariant polynomials are the constants.

Answer 2

Expanding on a remark by Maik Pickl:

Assume $p(x,y,z)$ is a non constant polynomial. Then there exist at least two points, $(x_1,y_1,z_1),(x_2,y_2,z_2) \in \Bbb{C}^3$, (with $x_1,y_1,z_1,x_2,y_2,z_2 \neq 0$), such that $$ p(x_1,y_1,z_1) \neq p(x_2,y_2,z_2). $$ Let $\tau \in GL_3(\Bbb{C})$ be defined by $$ \tau(x,y,z) := (\frac{x_2}{x_1} x,\frac{y_2}{y_1} y,\frac{z_2}{z_1} z).$$ Now $$p\circ \tau (x_1,y_1,z_1) = p(x_2,y_2,z_2) \neq p(x_1,y_1,z_1),$$ hence $p$ is not $\tau$-invariant, and cannot be $GL_3(\Bbb{C})$-invariant.