| #### 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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