Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^\*$.


Assuming I can find polynomials $f\_1, \dots, f\_n \colon \mathfrak{g}^\* \to \mathbb{R}$, such that $f\_1 \circ \Phi, \dots, f\_n \circ \Phi \colon M \to \mathbb{R}$ are functionally independent in at least one point in $M$. Is it now possible to conclude, that $\Phi^\*f\_1, \dots \Phi^\*f\_n$ are functionally independent on some open and dense subset in $M$? Are the functions $\Phi^\*f\_j$ for instance real analytic?



Polynomials pulled back by momentum maps
If group, action, manifold, and symplectic form $\omega$ are real analytic then also the momentum map is real analytic. The reason for this is that the Poincar\'e lemma also works in the real analytic setting. More precisely, let $\xi\in\mathfrak g$. Then (at least locally) there is a real analytic function $f\_\xi$ on $M$ with $df\_\xi=\omega(\xi x,\ast)$. The component function $\Phi\_\xi(x):=\langle\Phi(x),\xi\rangle$ has the same property, so it differs from $f\_\xi$ by a constant.


