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D.8.7.15 num_elim

Procedure from library recover.lib (see recover_lib).

Usage:
num_elim(I,f,D); ideal I, poly f, int D
f the product of the ring variables you want to eliminate D a bound to the degree of the elements of the components

Return:
ideal: the ideal obtained from I by eliminating the variables specified in f

Remarks:
This procedure uses Bertini to compute a set of witness points for I, projects them onto the components corresponding to the variables specified in f and then proceeds as num_radical_via_randlincom.

Note:
Should only be called from a ring over the rational numbers.

Example:
 
LIB "recover.lib";
ring r=0,(x,y,z),dp;
poly f1=x-y;
poly f2=z*(x+3y);
poly f3=z*(x2+y2);
ideal I=f1,f2,f3;
//First, we attempt to compute the elimination ideal with the hybrid algorithm.
ideal E1=num_elim(I,z,3,200);
==> 
==>    Bertini(TM) v1.6
==>     (May 22, 2018)
==> 
==>  D.J. Bates, J.D. Hauenstein,
==>  A.J. Sommese, C.W. Wampler
==> 
==> (using GMP v6.0.0, MPFR v3.1.2)
==> 
==> 
==> 
==> NOTE: You have requested to use adaptive path tracking.  Please make sure\
    that you have
==> setup the following tolerances appropriately:
==> CoeffBound: 4.000000000000e+00, DegreeBound: 3.000000000000e+00
==> AMPSafetyDigits1: 1, AMPSafetyDigits2: 1, AMPMaxPrec: 672
==> 
==> 
==> Tracking regeneration codim 1 of 2: 3 paths to track.
==> Tracking path 0 of 3
==> Tracking path 1 of 3
==> Tracking path 2 of 3
==> 
==> Sorting codimension 1 of 2: 3 paths to sort.
==> Sorting 0 of 3
==> Sorting 1 of 3
==> Sorting 2 of 3
==> 
==> Preparing regeneration codim 2 of 2: 3 witness points to move.
==> Moving 0 of 3
==> Moving 1 of 3
==> Moving 2 of 3
==> 
==> Tracking regeneration codim 2 of 2: 6 paths to track.
==> Tracking path 0 of 6
==> Tracking path 1 of 6
==> Tracking path 2 of 6
==> Tracking path 3 of 6
==> Tracking path 4 of 6
==> Tracking path 5 of 6
==> 
==> Sorting codimension 2 of 2: 6 paths to sort.
==> Sorting 0 of 6
==> Sorting 1 of 6
==> Sorting 2 of 6
==> Sorting 3 of 6
==> Sorting 4 of 6
==> Sorting 5 of 6
==> 
==> 
==> ************ Regenerative Cascade Summary ************
==> 
==> NOTE: nonsingular vs singular is based on rank deficiency and identical e\
   ndpoints
==> 
==> |codim|   paths   |witness superset| nonsingular | singular |nonsolutions\
   | inf endpoints | other bad endpoints
==> ----------------------------------------------------------------------------------------------------------------
==> | 1   |   3       |   0            |  0          |  0       |  3         \
   |   0           |  0
==> | 2   |   6       |   2            |  2          |  0       |  3         \
   |   1           |  0
==> ----------------------------------------------------------------------------------------------------------------
==> |total|   9
==> 
==> ****************************************************
==> 
==> 
==> 
==> *************** Witness Set Summary ****************
==> 
==> NOTE: nonsingular vs singular is based on rank deficiency and identical e\
   ndpoints
==> 
==> |codim| witness points | nonsingular | singular 
==> -------------------------------------------------
==> | 2   |   2            |  2          |  0       
==> -------------------------------------------------
==> 
==> ****************************************************
==> 
==> 
==> Calculating traces for codimension 2.
==> Calculating 0 of 2
==> Calculating 1 of 2
==> 
==> 
==> ************* Witness Set Decomposition *************
==> 
==> | dimension | components | classified | unclassified
==> -----------------------------------------------------
==> |   0       |   2        |   2        |  0
==> -----------------------------------------------------
==> 
==> ************** Decomposition by Degree **************
==> 
==> Dimension 0: 2 classified components
==> -----------------------------------------------------
==>    degree 1: 2 components
==> 
==> *****************************************************
==> 
==> 0
//Now, we compute the elimination ideal symbolically.
ideal E2=elim(I,z);
//Lastly, we compare the results.
E1;
==> E1[1]=0
E2;
==> E2[1]=x-y


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