A factorization theorem for unfoldings of analytic functions

Abstract:

Let $\tilde f$ and $g$ be holomorphic function germs at 0 in ${{\mathbf {C}}^n} \times {{\mathbf {C}}^n} = \left \{ {\left ( {x,s} \right )} \right \}$. If ${d_x}g\Lambda {d_x}\tilde f = 0$ and if $f\left ( x \right ) = \tilde f\left ( {x,0} \right )$ is not a power or a unit, then there exists a germ $\lambda$ at 0 in ${{\mathbf {C}}^n} \times {{\mathbf {C}}^n}$ such that $g\left ( {x,s} \right ) = \lambda \left ( {\tilde f\left ( {x,s} \right ),s} \right )$. The result has the implication that the notion of an RL-morphism in the unfolding theory of foliation germs generalizes that of a right-left morphism in the function germ case.