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D.5.1.9 paraConic
Procedure from library paraplanecurves.lib (see paraplanecurves_lib).
- Usage:
- paraConic(q); q poly
- Assume:
- The basering must be a polynomial ring in three variables with
coefficients in Q.
The polynomial q must be homogeneous of degree 2 and absolutely
irreducible.
- Note:
- The procedure might fail or give a wrong output if the assumptions
do not hold.
- Return:
- ring with an ideal PARACONIC. The ring should be considered as the
homogeneous coordinate ring of PP^1, the ideal defines a rational
parametrization PP^1 --> C2 = {q=0}.
- Theory:
- We compute a point on C2 via rationalPointConic. The pencil of
lines through this point projects C2 birationally to PP^1. Inverting
the projection gives the result.
Example:
| | LIB "paraplanecurves.lib";
ring R = 0,(x,y,z),dp;
poly f = y^8-x^3*(z+x)^5;
ideal adj = adjointIdeal(f);
def Rn = invertBirMap(adj,ideal(f));
==> // 'invertBirMap' created a ring together with two ideals J and psi.
==> // Supposing you typed, say, def RPn = invertBirMap(phi,I);
==> // you may access the ideals by typing
==> // setring RPn; J; psi;
setring(Rn);
J;
==> J[1]=y(5)*y(6)-y(4)*y(7)
==> J[2]=y(4)*y(6)-y(3)*y(7)
==> J[3]=y(2)*y(6)-y(1)*y(7)
==> J[4]=y(4)*y(5)-y(2)*y(7)
==> J[5]=y(3)*y(5)-y(1)*y(7)
==> J[6]=y(1)*y(5)-y(7)^2
==> J[7]=y(4)^2-y(1)*y(7)
==> J[8]=y(3)*y(4)-y(1)*y(6)
==> J[9]=y(2)*y(4)-y(7)^2
==> J[10]=y(1)*y(4)-y(6)*y(7)
==> J[11]=y(2)*y(3)-y(6)*y(7)
==> J[12]=y(1)*y(3)-y(6)^2
==> J[13]=y(2)^2-y(5)*y(7)
==> J[14]=y(1)*y(2)-y(4)*y(7)
==> J[15]=y(1)^2-y(3)*y(7)
==> J[16]=y(1)*y(6)^2-y(3)^2*y(7)
==> J[17]=y(6)^4-y(3)^3*y(7)
def Rc = rncItProjEven(J);
PHI;
==> PHI[1]=-y(7)
==> PHI[2]=-y(2)
==> PHI[3]=-y(5)
setring Rc;
CONIC;
==> y(2)^2-y(1)*y(3)
def RPc = paraConic(CONIC);
==> // 'paraConic' created a ring together with an ideal RNC.
==> // Supposing you typed, say, def RP1 = paraConic(q);
==> // you may access the ideal by typing
==> // setring RP1; PARACONIC;
setring RPc;
PARACONIC;
==> PARACONIC[1]=s2
==> PARACONIC[2]=st
==> PARACONIC[3]=t2
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See also:
rationalPointConic.
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