algcurves
parametrization
find a parametrization for a curve with genus 0
Calling Sequence
Parameters
Description
Examples
parametrization(f, x, y, t)
f
-
irreducible polynomial in x and y, with genus 0
x, y, t
variables
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 Xt,Yt of rational functions in t, such that Xt,Yt 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.
withalgcurves:
f≔y5+2xy2+2xy3+x2y−4x3y+2x5:
v≔parametrizationf,x,y,t
v≔646272t5+132192t4+6120t3−238t2−17t5430596t5−103680t4−17280t3−1440t2−60t−1,−2594064t5−98260t4+9826t35430596t5−103680t4−17280t3−1440t2−60t−1
Now subs(t=any number,v) should be a point on the curve. Test the result (this should be 0):
normalsubsx=v1,y=v2,f
0
parametrizationx4+y4+ax2y2+by3,x,y,t
−bt3t4+at2+1,−t4bt4+at2+1
See Also
AFactor
algcurves[genus]
algcurves[Weierstrassform]
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