= Rational interpolation linearization
{title2=$q(x_i)=f_ir(x_i)$}
For <rational interpolation> with $\deg q\leq m$ and a monic denominator $r(x)=x^\ell-\sum_{j=0}^{\ell-1}b_jx^j$, multiplying through gives the <linear system>
$$
\sum_{j=0}^{m}a_jx_i^j+f_i\sum_{j=0}^{\ell-1}b_jx_i^j=f_ix_i^\ell.
$$
These 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 $(1,x_i,\ldots,x_i^n)$, 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 $W=x^{m+1}R+S$, $\deg S\leq m$. 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>.
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