Rational interpolation linearization ($q(x_i)=f_{i}r(x_i)$)

For rational interpolation with $\mathrm{deg}q\le m$ and a monic denominator $r(x)=x^{\ell }-{\sum }_{j=0}^{\ell -1}{b}_j{x}^j$, multiplying through gives the linear systemThese equations are equivalent to the fraction conditions only when all $r(x_i)$ are nonzero. If $f_i=x_i^{m+1}$ and $m+\ell =n$, the coefficient matrix is the Vandermonde matrix with rows $(1,x_i,\dots ,x_i^n)$, so distinct interpolation nodes give a unique coefficient solution. More constructively, divide $W(x)={\prod }_{i=0}^n(x-x_i)$ by $x^{m+1}$ as $W=x^{m+1}R+S$, $\mathrm{deg}S\le m$. Then $r=R$, $q=-S$, since $x^{m+1}r-q$ is the unique monic degree-$n+1$ polynomial vanishing at all the interpolation nodes. The quotient may vanish at a node, so unique solvability of this linear system does not prove existence of a rational interpolant.