Let $\sigma : \Bbb{C}^3 \to \Bbb{C}^3$ be a polynomial mapping.
Let $P:= \Bbb{C}[x,y,z]$ denote the space of polynomial in 3 variables.
Then $\sigma$ induces a (linear) mapping $\tilde{\sigma} : P\to P$, by $$p \mapsto \tilde{\sigma}(p) := p\circ \sigma,$$ for $p \in P$.
A fixed point for $\sigma$ is a point $(a,b,c)\in \Bbb{C}^3$, such that $\sigma(a,b,c) = (a,b,c)$, and similarly for $\tilde{\sigma}$.
Question: What is the relation, if any, between fixed points for $\sigma$ and for $\tilde{\sigma}$ ?