| #### D.8.7.9 num_prime_decom Procedure from library `recover.lib` (see recover_lib). Usage: num_prime_decom(I,D); ideal I, int D D a bound to the degree of the elements of the components of a prime decomposition of I. Return: list of ideals: each of the ideals a prime component of the radical of I Remarks: Uses Bertini. Note: Should only be called from a ring over the rational numbers. Example:
    LIB "recover.lib";
    ring R=0,(x,y,z),dp;
    ideal I=(x+y)*(y+2z), (x+y)*(x-3z);
    int D=2;
    int Prec=300;
    num_prime_decom(I,D,Prec);
    ==>
    ==>    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: 8.000000000000e+00, DegreeBound: 2.000000000000e+00
    ==> AMPSafetyDigits1: 1, AMPSafetyDigits2: 1, AMPMaxPrec: 1024
    ==>
    ==>
    ==> Tracking regeneration codim 1 of 2: 2 paths to track.
    ==> Tracking path 0 of 2
    ==> Tracking path 1 of 2
    ==>
    ==> Sorting codimension 1 of 2: 2 paths to sort.
    ==> Sorting 0 of 2
    ==> Sorting 1 of 2
    ==>
    ==> Preparing regeneration codim 2 of 2: 1 witness point to move.
    ==> Moving 0 of 1
    ==>
    ==> Tracking regeneration codim 2 of 2: 2 paths to track.
    ==> Tracking path 0 of 2
    ==> Tracking path 1 of 2
    ==>
    ==> Sorting codimension 2 of 2: 2 paths to sort.
    ==> Sorting 0 of 2
    ==> Sorting 1 of 2
    ==>
    ==>
    ==> ************ 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   |   2       |   1            |  1          |  0       |  1         \
       |   0           |  0
    ==> | 2   |   2       |   2            |  1          |  1       |  0         \
       |   0           |  0
    ==> ----------------------------------------------------------------------------------------------------------------
    ==> |total|   4
    ==>
    ==> ****************************************************
    ==>
    ==>
    ==> Removing junk points from codimension 2: 1 endpoints to check.
    ==> Checking 0 of 1
    ==>
    ==>
    ==> *************** Witness Set Summary ****************
    ==>
    ==> NOTE: nonsingular vs singular is based on rank deficiency and identical e\
       ndpoints
    ==>
    ==> |codim| witness points | nonsingular | singular
    ==> -------------------------------------------------
    ==> | 1   |   1            |  1          |  0
    ==> | 2   |   1            |  1          |  0
    ==> -------------------------------------------------
    ==>
    ==> ****************************************************
    ==>
    ==>
    ==> Calculating traces for codimension 1.
    ==> Calculating 0 of 1
    ==>
    ==> Using combinatorial trace test to decompose codimension 1.
    ==>
    ==> Calculating traces for codimension 2.
    ==> Calculating 0 of 1
    ==>
    ==>
    ==> ************* Witness Set Decomposition *************
    ==>
    ==> | dimension | components | classified | unclassified
    ==> -----------------------------------------------------
    ==> |   1       |   0        |   0        |  1
    ==> |   0       |   1        |   1        |  0
    ==> -----------------------------------------------------
    ==>
    ==> ************** Decomposition by Degree **************
    ==>
    ==> Dimension 0: 1 classified component
    ==> -----------------------------------------------------
    ==>    degree 1: 1 component
    ==>
    ==> *****************************************************
    ==>
    ==> 0
    ==> empty list
    //Let us compare that to the result of primdecSY:
    primdecSY(I);
    ==> [1]:
    ==>    [1]:
    ==>       _[1]=x+y
    ==>    [2]:
    ==>       _[1]=x+y
    ==> [2]:
    ==>    [1]:
    ==>       _[1]=y+2z
    ==>       _[2]=x-3z
    ==>    [2]:
    ==>       _[1]=y+2z
    ==>       _[2]=x-3z

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