Assume that $p^*,q^*\in\mathbb R[x]$ are coprime, $\deg p^*=m$, $\deg q^*=n$, and $q^*(x)>0$ for every $x\in[a,b]$. Assume also that the linearised numerator space
has dimension $m+n+1$ and is a Haar space on $[a,b]$. Then $r^*$ is characterised among sufficiently small type $(m,n)$ rational perturbations by the existence of points $a\le x_0<x_1<\cdots <x_{m+n+1}\le b$ such that