Example 5.5 (A nonreduced scheme)

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Software
The computations described below were performed using Bertini version 1.4 and Matlab version R2012b in a 64-bit Linux environment. The instructions below assume that the user already has these or similar versions installed. Note that the Bertini commands are written here to run in serial, but it would be advisable to run these commands in parallel if the computing environment is set up for it (e.g. mpirun bertini input instead of bertini input).

Part 1: The algebraic set V

Download Example5_5.zip and extract the files in an empty directory.
At the command line, run:
bertini input.sliceC
cp witness_data witness_data_copy
At the command line, run:
bertini input.sampleC
Follow the on screen prompts to sample 5000 points on the first component listed with dim 1, deg 8. Name the file sample_c1. Run this step a second time to sample 5000 points on the second component of dim 1, deg 8. Name the file sample_c2. Note: The results for this example as an algebraic set could be obtained with fewer (e.g. 1000) points on each component, but here we sample 5000 each because we will need them for the scheme computations later.
In Matlab, run:
combinepoints(['sample_c1';'sample_c2'],'sample_c3');
split_witness_data;
In Matlab, run:
testacm
When prompted, answer with:
w1, sample_c1, 8
The script may give a warning when attempting to compute HP. A Bertini input file will be generated which can be used to finish this computation.
At the command line, run:
bertini input.hp
The coefficients of HP_X can now be read from finite_solutions. These coefficients can be converted to rational numbers, e.g., using the Maple command: convert(...,rational,80)
Repeat the previous step, but when prompted answer with:
w2, sample_c2, 8
Repeat again, but when prompted answer with:
w3, sample_c3, 8

Part 2: The projective scheme S

In Matlab, run:
H = zeros(1,5); for ind=0:length(H)-1; [H(1,ind+1),S]=schemehfc('sample_c1',ind,1); end; disp(H);
HF for C1 will be shown: 1,7,20,36,52
In Matlab, run:
H = zeros(1,5); for ind=0:length(H)-1; [H(1,ind+1),S]=schemehfc('sample_c2',ind,1); end; disp(H);
HF for C2 will be shown: 1,7,20,36,52
In Matlab, run:
H = zeros(1,7); for ind=0:length(H)-1; [H(1,ind+1),S]=schemehfc('sample_c3',ind,0); end; disp(H);
Note: This step uses all 10000 sample points for simplicity, but the Matlab code could be modified to be more efficient.
HF for C3 will be shown: 1,7,23,49,80,112,144
At the command line, run:
cp witness_data_copy witness_data
bertini input.print
Answer the prompts to print data for the 1st component of dimension 1, degree 8. Call this file schemew1.
Repeat this step, but answer the prompts to print data for the 2nd component of dim 1, deg 8. Call this file schemew2.
In Matlab, run:
combinepoints(['schemew1';'schemew2'],'schemew3');
In Matlab, run:
H = zeros(1,5); for ind=0:length(H)-1; H(1,ind+1)=schemehfw('schemew1',ind); end; disp(H);
HF for W1 will be shown: 1,6,13,16,16
In Matlab, run:
H = zeros(1,5); for ind=0:length(H)-1; H(1,ind+1)=schemehfw('schemew2',ind); end; disp(H);
HF for W2 will be shown: 1,6,13,16,16
In Matlab, run:
H = zeros(1,7); for ind=0:length(H)-1; H(1,ind+1)=schemehfw('schemew3',ind); end; disp(H);
HF for W3 will be shown: 1,6,16,26,31,32,32

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