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D.8.7.14 num_radical2
Procedure from library recover.lib (see recover_lib).
- Usage:
- num_radical2(P,D,C); list P, int D, bigint C
P a list of lists representing a witness point set representing an ideal I
D should be a bound to the degree of the elements of the components
C the number with which the images of the Veronese embeddings are multiplied
- Return:
- list of ideals: each of the ideals a prime component of the radical of I
- Remarks:
- Instead of using the images of the Veronese embeddings of each individual witness
point, this procedure first computes a random linear combination of those images
and searches for homogeneous polynomial relations for this linear combination.
- Note:
- Should only be called from a ring over the complex numbers.
Example:
| | LIB "recover.lib";
//First, we write the input file for bertini and compute the radical symbolically.
ring r=0,(x,y,z),dp;
ideal I=4xy2-4z3,-2x2y+5xz2;
ideal Rad=radical(I);
writeBertiniInput(I,100);
//Then we attempt to compute the radical via the hybrid algorithm.
ring R=(complex,100,i),(x,y,z),dp;
system("sh","bertini input");
==>
==> 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: 3.000000000000e+00
==> AMPSafetyDigits1: 1, AMPSafetyDigits2: 1, AMPMaxPrec: 352
==>
==>
==> 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: 6 witness points to move.
==> Moving 0 of 6
==> Moving 1 of 6
==> Moving 2 of 6
==> Moving 3 of 6
==> Moving 4 of 6
==> Moving 5 of 6
==>
==> Tracking regeneration codim 2 of 2: 9 paths to track.
==> Tracking path 0 of 9
==> Tracking path 1 of 9
==> Tracking path 2 of 9
==> Tracking path 3 of 9
==> Tracking path 4 of 9
==> Tracking path 5 of 9
==> Tracking path 6 of 9
==> Tracking path 7 of 9
==> Tracking path 8 of 9
==>
==> Sorting codimension 2 of 2: 9 paths to sort.
==> Sorting 0 of 9
==> Sorting 1 of 9
==> Sorting 2 of 9
==> Sorting 3 of 9
==> Sorting 4 of 9
==> Sorting 5 of 9
==> Sorting 6 of 9
==> Sorting 7 of 9
==> Sorting 8 of 9
==>
==>
==> ************ 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 | 9 | 9 | 1 | 8 | 0 \
| 0 | 0
==> ----------------------------------------------------------------------------------------------------------------
==> |total| 12
==>
==> ****************************************************
==>
==>
==>
==> *************** Witness Set Summary ****************
==>
==> NOTE: nonsingular vs singular is based on rank deficiency and identical e\
ndpoints
==>
==> |codim| witness points | nonsingular | singular
==> -------------------------------------------------
==> | 2 | 3 | 1 | 2
==> -------------------------------------------------
==>
==> ****************************************************
==>
==>
==> Calculating traces for codimension 2.
==> Calculating 0 of 3
==> Calculating 1 of 3
==> Calculating 2 of 3
==>
==>
==> ************* Witness Set Decomposition *************
==>
==> | dimension | components | classified | unclassified
==> -----------------------------------------------------
==> | 0 | 3 | 3 | 0
==> -----------------------------------------------------
==>
==> ************** Decomposition by Degree **************
==>
==> Dimension 0: 3 classified components
==> -----------------------------------------------------
==> degree 1: 3 components
==>
==> *****************************************************
==>
==> Witness Points Deflated: 2
==> 0
list P=getWitnessSet();
int D=2;
bigint C=bigint(10)**30;
ideal Rad2=num_radical2(P,D,C);
==> [1]:
==> _[1]=5*yz-2*z2
//Lastly, we compare the results.
Rad2;
==> Rad2[1]=5*yz-2*z2
ideal Rad=fetch(r,Rad);
Rad;
==> Rad[1]=5*yz-2*z2
==> Rad[2]=4*xz-25*z2
==> Rad[3]=2*xy-5*z2
reduce(Rad,std(Rad2));
==> // ** groebner base computations with inexact coefficients can not be tru\
sted due to rounding errors
==> _[1]=0
==> _[2]=4*xz-25*z2
==> _[3]=2*xy-5*z2
reduce(Rad2,std(Rad));
==> // ** groebner base computations with inexact coefficients can not be tru\
sted due to rounding errors
==> _[1]=0
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See also:
num_radical1.
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