Question

The following assertion appears plausible to me: Let $f(z,w,u)$ and $g(z,w,u)$ be holomorphic in $z$, $w$, and $u$. Suppose that $f(z_0,w_0,u_0) = g(z_0,w_0,u_0) = 0$ and that $f$ and $g$ are non-degenerate in the sense that none of $f(z,w_0,u_0)$, $f(z_0,w,u_0)$,...,$g(z_0,w_0,u)$ are identically zero. Then there exists $w(u)$ and $z(u)$ defined near $u_0$ such that $f(z(u),w(u),u) = g(z(u),w(u),u) = 0$. Note that I am not imposing any uniqueness or regularity requirements on $z(u)$ and $w(u)$.

Is this true? What is a good reference for this?

Answer 1

In general, $S:={(z,w,u):f=g}$ will have codimension 1. If we arrange for $S$ to be contained entirely in the fiber over $u_0$, then the desired functions $z(u)$, $w(u)$ won't exist.

In particular, if fix $z_0$, $w_0$, find $h(z,w)$ holomorphic with $h(z_0,w_0)=0$, then we can take $u_0=0$, $f(z,w,u)=h(z,w)+u$, $g(z,w,u)=h(z,w)+2u$ (we will need some condition on $h$ so that $f$ and $g$ are non-degenerate).