The purpose of this routine is to reconstruct a rational function $\frac{n}{d}$ in x from its image $u\mathbf{mod}m$ where u and m are polynomials in $F_x$, and $F$ is a field of characteristic 0. Given positive integers N and D, ratrecon returns the unique rational function $r=\frac{n}{d}$ if it exists satisfying $r=u\mathbf{mod}m$, $\mathrm{degree}\left(n\,x\right)\le N$, $\mathrm{degree}\left(d\,x\right)\le \mathrm{D}$, and $\mathrm{lcoeff}\left(d\,x\right)=1$. Otherwise ratrecon returns FAIL, indicating that no such polynomials n and d exist.  The rational function r exists and is unique up to multiplication by a constant in $F$ provided the following conditions hold:

If the integers N and D are not specified, they both default to be the integer $\mathrm{floor}\left(\frac{\left(\mathrm{degree}\left(m,x\right)-1\right)}{2}\right))$.

Note, in order to use this routine to reconstruct a rational function $r=\frac{n}{d}$ from u satisfying $r=u\mathbf{mod}m$, the modulus m being used must be chosen to be relatively prime to d. Otherwise the reconstruction returns FAIL.

The special case of $m=x^k$ corresponds to computing the N,D Pade approximate to the series u of order $\mathrm{O}\left(x^k\right)$.

For the special case of $N=0$, the polynomial $\frac{d}{n}$ is the inverse of u in $\frac{F_x}{m}$ provided u and m are relatively prime.

$s≔\mathrm{convert}\left(\mathrm{series}\left(\exp \left(x\right)\,x\right)\,\mathrm{polynom}\right)$

$s≔1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\frac{1}{24}{x}^4+\frac{1}{120}{x}^5$

$\mathrm{ratrecon}\left(s\,x^6\,x\,3\,2\right)$

$\frac{20+\frac{1}{3}{x}^3+3x^2+12x}{x^2-8x+20}$

$\mathrm{ratrecon}\left(s\,x^6\,x\,2\,3\right)$

$\frac{-3x^2-24x-60}{x^3-9x^2+36x-60}$

$\mathrm{ratrecon}\left(s\,x^6\,x\,3\,3\right)$

Error, (in ratrecon) degree bounds too big

$\mathrm{ratrecon}\left(x^2+1\,x^3\,x\,1\,1\right)$

$\mathrm{FAIL}$

$r≔\mathrm{ratrecon}\left(x-1\,x^3-2\,x\,0\,2\right)$

$r≔\frac{1}{x^2+x+1}$

$\mathrm{rem}\left(\frac{x-1}{r}\,x^3-2\,x\right)$

$1$