I'm working through some lecture notes and I get stuck to understand the following:
Let G be a compact Lie group. Let $p: M \rightarrow B $ be a G-equivariat real vector bundle on a compact G-manifold $B$. Let $\lambda $ be a G-invariant 1-form on $ M$ such that $\lbrace f_\lambda = 0 \rbrace = B $ where $f_\lambda : M \rightarrow \mathfrak{g}^*$ is the G-equivariat map defined by
$< f_\lambda(m) ,X)>:= \lambda_m(X_M(m))$, $\quad m\in M , X\in \mathfrak{g}.$
Suppose that the 1-form $\lambda$ is homogeneous on the fibers of M, with strictly positive degree of homogeneity . Then each component of the differential form $(t,m) \rightarrow e^{-itd\lambda_m}\lambda_m$ ( where $t \in \mathbb{R}$) is bounded by a function of the form $P(t,||m||)$, where P is a polynomial in both variables.
What do they mean by saying "each component of the differential form", and why there exists the polynomial P ?