This procedure computes, if it exists, a parametrization of an algebraic curve f. A parametrization is a birational equivalence from a projective line to the given curve f. Such a parametrization exists if and only if the genus is 0 and the curve is irreducible (which can be checked by AIrreduc).

The output of the procedure is a list $\left[X\left(t\right)\,Y\left(t\right)\right]$ of rational functions in t, such that $\left[X\left(t\right)\,Y\left(t\right)\right]$ is a point on the curve f for every value of t.

For a description of the method used see M. van Hoeij, "Rational Parametrizations of Algebraic Curves using a Canonical Divisor", 23, p. 209-227, JSC 1997.

$\mathrm{with}\left(\mathrm{algcurves}\right)\:$

$f≔y^5+2x{y}^2+2x{y}^3+x^{2}y-4x^{3}y+2x^5\:$

$v≔\mathrm{parametrization}\left(f\,x\,y\,t\right)$

$v≔\left[\frac{-24192t^5-6048t^4+2520t^3-238t^2+7t}{181604t^5-103680t^4+17280t^3-1440t^2+60t-1}\,\frac{16464t^5+6860t^4-686t^3}{181604t^5-103680t^4+17280t^3-1440t^2+60t-1}\right]$

$\mathrm{normal}\left(\mathrm{subs}\left(x=v\left[1\right]\,y=v\left[2\right]\,f\right)\right)$

$0$

$\mathrm{parametrization}\left(x^4+y^4+a{x}^2{y}^2+b{y}^3\,x\,y\,t\right)$

$\left[-\frac{b{t}^3}{t^4+a{t}^2+1}\,-\frac{t^{4}b}{t^4+a{t}^2+1}\right]$