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D.5.1.2 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:
 


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