projectionToHypersurface -- Generic projection to a hypersurface

Synopsis

Usage:

projectionToHypersurface I

projectionToHypersurface R
Inputs:
- I, an ideal, an ideal in a polynomial ring
- R, a ring, a quotient of a polynomial ring
Optional inputs:
- Codimension => an integer, default value null, specified if you already know the codimension of your Ideal (or QuotientRing) in your ambient ring
- MaxCoordinatesToReplace => an integer, default value infinity, to be passed to randomCoordinateChange
- Replacement => a symbol, default value Binomial, to be passed to randomCoordinateChange
- Homogeneous => a Boolean value, default value true, to be passed to randomCoordinateChange
- Verbose => a Boolean value, default value false, set to true for verbose output
Outputs:
- a ring map, a list with two entries, the generic projection map, and the ideal if I was provided, or the ring if R was provided

Description

This creates a projection to a hypersurface. It differs from genericProjection(codim I - 1, I) as it only tries to find a hypersurface equation that vanishes along the projection, instead of finding one that vanishes exactly at the projection. This can be faster, and can be useful for finding points.

i1 : R=ZZ/5[x,y,z];

i2 : I = ideal(random(3,R)-2, random(2,R));

o2 : Ideal of R

i3 : projectionToHypersurface(I)

              ZZ                             6     5      4 2       5    6  
o3 = (map (R, --[y..z], {x, y - 2z}), ideal(y  - 2y z - 2y z  - 2y*z  + z  -
               5
     ------------------------------------------------------------------------
       2       2
     2y z + y*z  - 1))

o3 : Sequence

i4 : projectionToHypersurface(R/I)

                                                                             
                                                                             
                                                   R                         
o4 = (map (------------------------------------------------------------------
                3       2     3    2     2        2       2     3            
           (- 2x  - 2x*y  - 2y  + x z - y z + 2x*z  - 2y*z  - 2z  - 2, - 2x*z
     ------------------------------------------------------------------------
                                               ZZ                            
                                               --[y..z]                      
                                                5                            
     --------------, --------------------------------------------------------
                 2    6    5      4 2     2 4      5     3    2       2    3 
      + 2y*z - 2z )  y  - y z + 2y z  - 2y z  - y*z  + 2y  - y z - y*z  + z  
     ------------------------------------------------------------------------
                                                      ZZ
                                                      --[y..z]
                                                       5
     ---, {y + 2z, x + z}), -------------------------------------------------
                             6    5      4 2     2 4      5     3    2       
     + 1                    y  - y z + 2y z  - 2y z  - y*z  + 2y  - y z - y*z
     ------------------------------------------------------------------------
     ----------)
     2    3
       + z  + 1

o4 : Sequence

If you already know the codimension is c, you can set Codimension=>c so the function does not compute it.

Ways to use projectionToHypersurface :

projectionToHypersurface(Ideal)
projectionToHypersurface(Ring)

For the programmer

The object projectionToHypersurface is a method function with options.

projectionToHypersurface -- Generic projection to a hypersurface

Synopsis

Description

See also

Ways to use projectionToHypersurface :

For the programmer