If $v$ vanishes at any boundary point $a$ on $\mathbb{R}\setminus\{p_{1},\ldots p_{k}\}$, or if $v$ vanishes at $p_{i}$ beyond minimal order, then $\frac{v}{z-a}$ (resp. $\frac{v}{z-p_{i}}$) is also a first order deformation with the same $TL$ boundary condition, subject to the growth constraints at infinity. Since the kernel dimension is finite, the divisions can only happen a finite number of times, producing the polynomial $f$. ∎