There exists $k$ monomials $t^{-v(j_{1})}z^{j_{1}},\dots,t^{-v(j_{k})}z^{j_{k}}$ that dominate $f_{t}$ in a neighborhood of $F$. Namely, any other monomial evaluated at a point near $F$ has a smaller order by $t$. Furthermore, the hypersurface

$$\sum\limits_{m=1}^{k}t^{-v(j_{m})}z^{j_{m}}=0$$

is isomorphic to the hyperplane $z_{1}+\dots+z_{k-1}+1=0$ under the multiplicative change of coordinates by an element of $SL_{n+1}(\mathbb{Z})$.