Find a parametrization for a curve with genus 0 - Maple Help

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algcurves[parametrization] - Find a parametrization for a curve with genus 0

Calling Sequence

parametrization(f, x, y, t)

Parameters

f

-

irreducible polynomial in x and y, with genus 0

x, y, t

-

variables

Description

• 

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.

Examples

withalgcurves:

f:=y5+2xy2+2xy3+x2y4x3y+2x5:

v:=parametrizationf,x,y,t

v:=13500t5+8775t4+2070t3+212t2+8t107897t5+71590t4+18920t3+2480t2+160t+4,1920t51696t4552t378t24t107897t5+71590t4+18920t3+2480t2+160t+4

(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

(2)

parametrizationx4+y4+ax2y2+by3,x,y,t

bt3t4+at2+1,t4bt4+at2+1

(3)

See Also

AFactor, algcurves[genus], algcurves[Weierstrassform]


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