#### D.5.11.2 invertBirMap

Procedure from library `paraplanecurves.lib` (see paraplanecurves_lib).

Usage:
  invertBirMap(phi, I); phi ideal, I ideal

Assume:
  The ideal phi in the basering R represents a birational map of the variety given by the ideal I in R to its image in projective space P = PP^(size(phi)-1).

Note:
  The procedure might fail or give a wrong output if phi does not define a birational map.

Return:
  ring, the coordinate ring of P, with an ideal named J and an ideal named psi.
  The ideal J defines the image of phi.
  The ideal psi gives the inverse of phi.
  Note that the entries of psi should be considered as representatives of classes in the quotient ring R/J.

Theory:
  We compute the ideal I(G) in R**S of the graph G of phi.
  The ideal J is given by the intersection of I(G) with S.
  The map psi is given by a relation mod J of those relations in I(G) which are linear in the variables of R.

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)
    psi;
    ==> psi[1]=-y(6)^2
    ==> psi[2]=-y(4)*y(7)
    ==> psi[3]=y(6)^2-y(5)*y(7)

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